Time series forecasting

Overview

Forecasting predicts future values from past ones. The cards on this page study seven settings, and the methods differ between them.

• One scalar series with a fixed horizon. Predict the next H values of a series from its last values. The deep benchmark, the online sequence memory and the Transformer survey work on this setting 2103.12057 §2 1512.05463 §5 2205.01138 §3. A gated recurrent model with k-means class inputs targets the same setting, on water demand 2512.06714 §3.2. A frozen 7B language model, reused one channel at a time for this setting, is in Time-LLM 2310.01728 §3. Features learned by contrastive pretraining, with a ridge regressor on top, target the same setting 2202.01575 §3 2202.01575 §4.1. A tuned LSTM, adjusted by a booster that uses three weaker forecasters, targets monthly campus electricity in the same way 2605.25305 §3.2.3.
• Many related series, with a distribution over futures. Predict the next values of several series jointly, and sample plausible futures rather than one point forecast 2101.12072 §1 2101.12072 §3. The benchmark pipeline also targets this multivariate setting 2310.06119 §4.1. Two newer cards also fit this setting: a dense MLP that takes covariates 2304.08424 §4, and a decoder-only model whose output is a mixture of distributions 2505.14766 §3.1.
• Many related series, point forecasts from MLP-style models. Predict the next values of every series in a set, with one point per step. The models here mix information across series and time with plain layers, and the graph between series is either absent or not required. STID adds a learned identity per series and per time slot 2208.05233 §3. TSMixer alternates time mixing and feature mixing 2303.06053 §4.1. FreTS works on the discrete Fourier transform of the input 2311.06184 §3.1. A single linear layer, with weights shared across series, is the baseline that long-horizon Transformer forecasters are compared against 2205.13504 §4.
• Many basins, each with fixed attributes. Train one network on the daily series of many entities, with each entity's static attributes as extra input. The hydrology cards use 531 river basins and 27 attributes 1907.08456 §2.4 1907.08456 §2.2.
• The path of an agent. Predict the future (x, y) positions of a pedestrian or a vehicle, usually with several plausible futures 2203.13777 §1 2006.14480 §4.
• A grid or graph of series. Predict the next value at every node of a network, or the next map of a radar grid or global weather field, from the recent history of all nodes 1712.04997 §5 2007.04417 §4.1 2212.12794 §GraphCast 2108.09091 §2. The spatial layout is used, so neighbouring series can inform each other. A neural-ODE model on road sensors uses the same layout, with a second graph built from traffic similarity 2106.12931 §4.1. STEP pretrains a masked Transformer on long traffic history and feeds its states into a learned sparse graph 2206.09113 §3.2. FourierGNN drops the station map altogether and builds one fully connected graph over the values of the window 2311.06190 §4.1. MSGNet learns no fixed layout at all: it learns one graph per period scale of the input 2401.00423 §4.3.
• Sparse, irregular records with a label. Each record has its own time, variable and value, and most variables are missing at any time. The task is usually classification. Forecasting serves as a pretext task before that classification 2107.14293 tab. 2 2107.14293 §3.2.6.
• A forecast used for another decision. The forecast is a means. It scores anomalies from its own prediction errors 2106.06947 §3.6, or it predicts a resilience curve that triggers interventions 2609.27903 §3.3 2609.27903 §3.4. Its accuracy is then only part of the result.

For the scalar setting, classical statistical models such as ETS and ARIMA need stationarity and one model per series, which makes them hard to scale to many series 2103.12057 §2. The deep models in the benchmark are instead trained as one global model per dataset, on windows drawn from all of its series 2103.12057 §3.1 2103.12057 §3.3.

Each series is scaled with statistics from its own training part. Training windows slide one step at a time, and each window pairs a past block of L = factor × H values with the next H values 2103.12057 §3.3 2103.12057 fig. 2. Every benchmark model ends in one dense layer with one neuron per horizon step 2103.12057 §3.2.1.

The evidence for the deep benchmark is one author-reported study: seven architectures on twelve public datasets with more than 50,000 series in total 2103.12057 §3.1 2103.12057 tab. 8. It runs no classical statistical baselines 2103.12057 §3.2. Its headline "best" numbers are chosen on the evaluation split, so the rankings are optimistic. Compare architectures by their mean and spread, not by single best cells 2103.12057 §4.1 2103.12057 fig. 3. The other cards use other data, metrics and protocols, so their numbers do not transfer to this benchmark (see Contradictions).

Variants

Each variant is a family of forecasters or models from one card. Scalar-series variants come first, then trajectory variants, then many-series, grid and graph variants, and last the variants whose forecast feeds another task.

Feed-forward MLP

Dense layers over the flattened window. In the benchmark it trained fastest and was the least accurate, and the paper flags it as the worst model 2103.12057 §4.2 2103.12057 §4.3. The released code stacks dense layers with no activation, so the MLP there is linear in its input. The MLP result therefore says little about non-linear MLPs 2103.12057 §3.2.1. The card for all benchmark variants is An Experimental Review on Deep Learning Architectures for Time Series Forecasting.

Recurrent cells: ERNN, LSTM, GRU, ESN

Recurrent cells read the window one step at a time. The grid varies one to four layers, 32 to 128 units, and whether the full sequence is returned 2103.12057 tab. 9. LSTM had the best single configuration on 4 of 12 datasets, the most of any recurrent cell 2103.12057 §4.1 2103.12057 tab. 11. The paper's text says fewer stacked layers did better, except for ESN, which did best with four layers and full sequences 2103.12057 §4.4.2. ESN had slow inference 2103.12057 §4.2. Background on the cell itself is on the recurrent neural network page.

1D convolutional network (CNN)

A stack of same-padded 1D convolutions, each with optional max pooling, then the dense output. The grid varies one, two or four layers, 16 to 64 filters, and pooling of 0 or 2 2103.12057 tab. 9. The four-layer model without pooling was best 2103.12057 §4.4.3 2103.12057 tab. 14. CNN ranked first on mean accuracy across configurations, and it had the smallest spread with LSTM 2103.12057 §4.1 2103.12057 §4.3. It was the most efficient deep model, and it gave the best time-accuracy balance 2103.12057 §4.2 2103.12057 fig. 5. Its single best cell was on Traffic-metr-la 2103.12057 tab. 11.

Temporal convolutional network (TCN)

Dilated causal convolutions with residual blocks. The grid varies layers, filters, dilations, kernel size and the return-sequence flag [arxiv:2103.12057 tab. 9]. The best TCN was a single layer with kernel 6 and 64 filters, returning only the last output 2103.12057 §4.4.3 [arxiv:2103.12057 tab. 15]. It was the best single cell on two datasets, the same count as ERNN and GRU 2103.12057 tab. 11. Its return-sequence result conflicts between the text and a table; see Contradictions. The TCN's own card is Temporal convolutional network.

Gated recurrent cell with k-means class inputs (DCGRU)

A Novel Deep Neural Network Architecture for Real-Time Water Demand Forecasting forecasts 15-minute demand for one district metering area. Each reading becomes a row with its value and four class features, where the classes come from k-means clustering on the demand values 2512.06714 §3.3 2512.06714 §4.1. A window of 96 such rows, the last 24 hours, passes through three per-row dense layers, then a 32-unit GRU and a one-unit GRU output 2512.06714 §3.4 2512.06714 tab. 5. The model has 4,187 parameters, against 30,403 for the GRUN baseline 2512.06714 tab. 6. The paper chose four classes from an elbow plot, and its distortion values are not tabulated 2512.06714 §4.1 fig. 3.

Author-reported, on two district metering areas with 2,500 test records each: one-step MAPE is 1.26% and 1.31%, against 2.06% and 2.46% for GRUN. The 96-step MAPE is 1.50% and 1.38%, against 4.33% and 4.96% 2512.06714 tab. 6. The EDCGRU variant inserts one virtual reading between each pair of real ones, which doubles the window to 192 rows 2512.06714 §3.5. It reaches 0.98% and 0.99% one-step MAPE, but trains more than four times longer, at 850 and 1,055 seconds against 192 and 234 for DCGRU 2512.06714 tab. 6. The loss and the input scaling are not stated 2512.06714 §3.6, and there is no official code 2512.06714 Paper vs code.

Tuned LSTM with a weaker-model booster (WSB)

Electricity Consumption Forecasting: An Approach Using Cooperative Ensemble Learning with SHapley Additive exPlanations forecasts 12 months of monthly electricity use at two campuses. It starts from 12 lagged consumption values, plus calendar dummies and climate variables, for 47 candidate predictors. Mean absolute SHAP values rank them, and the least important are removed one at a time while Random Forest and XGBoost are refitted. The lowest average MAE comes after 36 removals, which leaves 11 predictors 2605.25305 §3.2.1 2605.25305 tab. 5.

A genetic algorithm tunes the LSTM, Random Forest, SVR and XGBoost, using the mean MAE over five time-series folds as the objective. The paper reports that the genetic algorithm beat particle swarm optimisation for every model 2605.25305 §3.2.2. The booster sits on the tuned LSTM forecast. It subtracts the mean of the three weaker forecasts, scales the difference by a triangular weight over the 12 steps, and adds the result back 2605.25305 eq. 1 2605.25305 eq. 2 2605.25305 eq. 3. The global weight 0.0833 comes from a random search 2605.25305 §3.2.3.

Author-reported, on the 2023 test year at each campus, the boosted model has the lowest sMAPE in its table at Palmas: 13.90%, against 15.35% for the tuned LSTM alone and 21.82% for exponential smoothing 2605.25305 tab. 14. At Coronel Vivida the margin over the tuned LSTM is small, 18.72% against 18.80% sMAPE and 464.93 against 466.92 kWh MAE 2605.25305 tab. 15. The paper reports that every model is inaccurate around the 2023 peak at each campus 2605.25305 §4.1. Each search is 750 evaluations, and each evaluation is a five-fold fit, over three seeds 2605.25305 tab. 12 2605.25305 §3.2.2. The released code fits the SHAP ranking on the whole Palmas frame, including the test year 2605.25305 Paper vs code.

Multiscale MLP with seasonal and trend mixing (TimeMixer)

TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting uses only linear layers and GELU, with no attention 2405.14616 §3. The past is average-pooled by a factor of two, M times, so scale m has length P / 2^m 2405.14616 §3.1. Each scale is split into seasonal and trend parts with a moving average 2405.14616 §3.2 2405.14616 eq. 3. Seasonal parts are mixed from fine to coarse, and trend parts from coarse to fine 2405.14616 eq. 4 2405.14616 eq. 5. After L such blocks, each scale has its own linear predictor over time, and the forecast is their sum 2405.14616 §3.3 2405.14616 eq. 6.

Author-reported, at input length 96, it has the best MSE and MAE on all eight long-term benchmarks 2405.14616 tab. 2. On Weather its MSE is 0.240, against 0.265 for PatchTST 2405.14616 tab. 2. At input length 3072 it takes 0.016 s per iteration and 1411 MiB, against 0.094 s and 16119 MiB for PatchTST. The batch size is not stated 2405.14616 tab. 8. Removing the mixing across past scales raises M4 SMAPE from 11.723 to 12.468 2405.14616 tab. 5.

The paper does not mention normalisation. The official code wraps each scale in instance normalisation by default, and the paper's mixer width is not stated, so the code's choice is used 2405.14616 Paper vs code. Under per-model hyperparameter search, the gap to PatchTST shrinks 2405.14616 app. E. The mixing layers grow with input length, which the authors name as a limit for mobile use 2405.14616 app. J.

Dense encoder-decoder with covariates (TiDE)

Long-term Forecasting with TiDE: Time-series Dense Encoder is an MLP-only encoder-decoder. It has no attention, recurrence or convolution 2304.08424 §4. The encoder reads the past values, the past and future covariates and the static attributes of one series. A decoder then produces the whole horizon at once 2304.08424 §4.1 [arxiv:2304.08424 §4.2]. A temporal decoder adds each future step's known covariates, and a linear path from the look-back is added to the output, so a purely linear model is a special case 2304.08424 §4.2. The paper uses a look-back of 720 for all four horizons: 96, 192, 336 and 720 2304.08424 §5.1. Its per-dataset hidden widths range from 256 to 1024 2304.08424 tab. 8.

Author-reported, on Traffic the MSE is lower than PatchTST/64 at all four horizons, from 0.336 against 0.360 at horizon 96 to 0.386 against 0.432 at 720 2304.08424 tab. 2. On Weather it is behind PatchTST/64 at three of the four horizons 2304.08424 tab. 2. The paper states that inference is about five times faster than the best Transformer, but it gives no timing table 2304.08424 §1 2304.08424 §5.3. The released code adds a learned series-ID embedding that the paper does not describe, so a trained model needs the series index at inference 2304.08424 Paper vs code.

Time- and feature-mixing MLP (TSMixer)

TSMixer: An All-MLP Architecture for Time Series Forecasting alternates a time-mixing layer, shared across variates, with a two-layer feature-mixing MLP, shared across time steps. It adds residuals and normalisation, and a final linear layer maps input length to horizon 2303.06053 §4.1. The paper motivates the design with a linear analysis. A linear map can reproduce a periodic series exactly, and a smooth trend has a bounded error. That result covers linear models, not TSMixer, and it assumes periodic or Lipschitz-smooth structure 2303.06053 §3 [arxiv:2303.06053 eq. 2].

Author-reported, with input length 512, TSMixer on seven long-term benchmarks is comparable to PatchTST and to its own time-mixing-only ablation. Feature mixing adds little on those benchmarks 2303.06053 §5.1. The paper's average MSE improvement over PatchTST is −1.53% 2303.06053 tab. 3. On M5, the version with static and known-future features, TSMixer-Ext, reaches test WRMSSE 0.640, against 0.670 for TFT 2303.06053 tab. 4. Without those features, TSMixer reaches 0.737, against 0.742 for Autoformer, a gap smaller than either standard deviation 2303.06053 tab. 4. The M5 model and the TSMixer-Ext code are not released 2303.06053 Paper vs code.

Frequency-domain MLP (FreTS)

Frequency-domain MLPs are More Effective Learners in Time Series Forecasting takes the discrete Fourier transform of the input and runs two complex-valued MLPs on it: one across channels and one across time steps. A feed-forward head then predicts the whole horizon in one step 2311.06184 §3.1 2311.06184 eq. 5. The paper argues that most of the signal's energy sits in a few frequency components, which the design uses [arxiv:2311.06184 §3.2] 2311.06184 fig. 1.

Author-reported, at horizon 12, FreTS has the lowest MAE on Solar (0.120) and on Traffic (0.011) in its short-term table 2311.06184 tab. 1. On the long-term table it has the lowest MAE on most rows. PatchTST is ahead on Traffic at horizons 48 and 96, and on Electricity at horizon 336 [arxiv:2311.06184 tab. 2]. Removing either learner raises the error on the ablation datasets 2311.06184 tab. 3. The released code runs only the temporal learner, and it uses diagonal frequency weights, so the channel-learner ablation cannot be reproduced from it 2311.06184 Paper vs code.

Online sequence memory (HTM)

The sequence memory of Hierarchical Temporal Memory, studied in Continuous online sequence learning with an unsupervised neural network model. Each input becomes a sparse binary code, with about 2% of 2048 columns active, and each column holds 32 cells 1512.05463 §3.2 1512.05463 tab. 1. Cells on dendritic segments predict the next element, and the synapses are updated after each element by a Hebbian-style rule 1512.05463 §3.3. The authors state that the rule is not derived from gradient descent, and there is no buffered training set 1512.05463 §3.3 1512.05463 §6.1. The model keeps several predictions open until later input picks one 1512.05463 §4.3.

Author-reported results: it predicts artificial high-order sequences perfectly up to order 100 1512.05463 §4.4. On NYC taxi counts, 2.5 hours ahead, its error is comparable to the best LSTM and lower than ARIMA, ELM and ESN 1512.05463 §5 1512.05463 fig. 10. After a 20% pattern change, its error falls back in about two weeks 1512.05463 fig. 11. Under temporal noise, it is about 0.5 accuracy, and LSTM is slightly higher 1512.05463 §4.5. The released code's defaults differ from Table 1, so implement the paper's values as written 1512.05463 tab. 1.

Transformers for time series

Transformers in Time-series Analysis: A Tutorial is a tutorial. It runs no experiments and proposes no model. It explains self-attention with a scaling by the square root of the key dimension 2205.01138 §2.2.2, multi-head attention 2205.01138 §2.3, the encoder and decoder blocks 2205.01138 §2.4, and sinusoidal positional encoding with base 10,000 in the text 2205.01138 §2.5.1. See the self-attention and positional encoding pages for the building blocks.

It then lists time-series changes to these blocks, and the performance statements come from cited works, not from this paper 2205.01138 §4:

• Informer uses ProbSparse attention and distilling. The cited work reports better results than ARIMA, Prophet and LSTM-based baselines on electricity load 2205.01138 §4.1.
• LogSparse reduces the memory of attention from O(L²) to O(L log₂ L) per layer 2205.01138 §4.2.
• SAnD uses a 1D convolution embedding and restricted self-attention 2205.01138 §4.3. The Traffic Transformer adds a graph network for spatial dependence and gives no numbers 2205.01138 §4.4.
• Influenza forecasting adds a time-delay embedding. The best embedding size is 8, with no RMSE values given 2205.01138 §4.6.

The paper gives almost no hyperparameters, and no official code is linked 2205.01138 §5.

Exponential-smoothing and Fourier attention (ETSformer)

ETSformer: Exponential Smoothing Transformers for Time-series Forecasting replaces the attention of a Transformer with two blocks borrowed from classical forecasting. Exponential smoothing attention weights past values with a decay that falls off with lag. Frequency attention keeps the strongest Fourier bases and carries them past the window 2202.01381 §4.2. Each encoder stack removes a season, a growth term and a level from its input. The decoder damps the growth, extends the season and adds both to the last level 2202.01381 §4.1 eq. 3. The authors report the lowest MSE on 22 of the 24 multivariate rows in their table, against Autoformer and Informer among others 2202.01381 tab. 1. Their ablation removes each block in turn, and every ablated variant is worse than the full model on the four datasets shown 2202.01381 tab. 2. The text claims 35 wins in 40 settings, which the table does not show, so quote the table 2202.01381 §5.1. The released code's smoothing parameter is one minus the paper's α, and it differs in other ways listed on the card 2202.01381 Paper vs code.

Linear models: LTSF-Linear, DLinear and NLinear

Are Transformers Effective for Time Series Forecasting? defines three one-layer linear forecasters. LTSF-Linear maps the last values of each series to all future values in one step 2205.13504 §4. DLinear adds a moving-average trend split and sums two such layers, one for the trend and one for the remainder 2205.13504 §4. NLinear subtracts the last observed value before its layer and adds it back afterwards, to reduce the shift between training and test periods 2205.13504 §4. All three share their weights across series, so they model no dependence between series 2205.13504 §4.

On nine public benchmarks, the best linear model has lower MSE than the best Transformer baseline in each of the 36 dataset-horizon cells of the paper's main table 2205.13504 tab. 2. The comparison is not like for like. The linear models use a look-back of 336 steps and the Transformers 96 2205.13504 app. B.2, and most Transformer numbers are copied from the FEDformer paper rather than re-run 2205.13504 tab. 2. Plain Linear is not always ahead: on ETTh2 at horizon 720 its MSE is 0.698, against 0.463 for FEDformer 2205.13504 tab. 2. The table gives no run count or spread 2205.13504 tab. 2.

The cost gap is large. On one Electricity setting, DLinear needs 0.04 GMACs and 139.7K parameters, against 4.03 GMACs and 13.61M parameters for a Transformer with the same decoder 2205.13504 tab. 8. The timings and memory use are for one GPU and one setting, so they do not generalise by themselves 2205.13504 tab. 8.

Contrastive seasonal-trend features with a ridge head (CoST)

CoST: Contrastive Learning of Disentangled Seasonal-Trend Representations for Time Series Forecasting learns a per-step representation first and forecasts afterwards. A dilated convolution encoder outputs 320 channels, split into a trend half and a seasonal half 2202.01575 §3 2202.01575 app. E. The trend half is trained with a MoCo-style contrastive loss over a mixture of causal convolutions of different kernel sizes 2202.01575 §3.1. The seasonal half passes through a learnable Fourier layer, and a contrastive loss on amplitudes and phases of its spectrum 2202.01575 §3.2. Because the seasonal loss works in the frequency domain, the period need not be known in advance 2202.01575 §1. After pretraining on the training split, the encoder is frozen and a ridge regression maps its features to the whole horizon 2202.01575 §4.1.

Author-reported, on five multivariate benchmarks, the average MSE is 0.590 against 0.750 for TS2Vec, the best feature-based baseline, which is 21.3% lower. It is 39.3% lower than an end-to-end TCN at 0.972 2202.01575 §4.2 2202.01575 tab. 1. In the univariate average, CoST reports 0.193 against 0.203 for TS2Vec 2202.01575 tab. 7. The ablation in 2202.01575 tab. 3 removes one component at a time on the ETT datasets. Training costs more than TS2Vec: 262.78 s against 91.9 s, on one A100. The table repeats these training times for every horizon 2202.01575 tab. 6 2202.01575 app. F. The paper blames running the trend experts one after another 2202.01575 app. F. Its Informer and LogTrans rows are copied from other papers, not re-run 2202.01575 app. E. The official code is salesforce/CoST, under BSD-3-Clause, but it differs from the paper in several places (see Contradictions) 2202.01575 Paper vs code. The card is CoST: Contrastive Learning of Disentangled Seasonal-Trend Representations for Time Series Forecasting.

Zero-shot tabular regression (TabPFN-TS)

From Tables to Time: Extending TabPFN-v2 to Time Series Forecasting casts a forecast as tabular regression 2501.02945 §3.1. Each time step is one row. The features are a running index, 17 calendar values, the top-k seasonal sine and cosine pairs from a spectral method, and any covariates known for the future 2501.02945 §3.1 2501.02945 alg. 1. The observed history is the training set. A pretrained TabPFN-v2 with 11M parameters fills in the future rows in one pass, with no time-series training 2501.02945 §1 2501.02945 §3.2. Each series is a separate forward pass 2501.02945 §6.2. The model uses the last 4096 observations as context 2501.02945 app..

Author-reported, and the per-task numbers appear only in figures. On 97 univariate GIFT-Eval tasks it ranks closely behind TiRex, Toto-1.0 and Moirai-2 2501.02945 §4.1. On 28 fev-bench tasks with known future covariates it is the strongest overall 2501.02945 §4.2. Covariates help most synthetic classes but not linear trends 2501.02945 §5.3. Its runtime is about 30 times that of Chronos-Bolt-Base and TimesFM-2.0 2501.02945 §6.2.

It does not extrapolate linear or exponential trends beyond the context range 2501.02945 §5.1. Only covariates known for the future are supported, such as holidays, and not unknown ones such as weather 2501.02945 §3.1. The released code pins a newer checkpoint (v3.5) than the paper's v2, and its default uses 12 seasonal periods where the paper uses 5. Reproducing the paper needs both changed back 2501.02945 Paper vs code. For the backbone itself, see the tabular foundation model page.

Decoder-only foundation model with causal scaling (Toto)

This Time is Different: An Observability Perspective on Time Series Foundation Models is a 151M-parameter decoder-only foundation model for observability metrics. It cuts each series into 64-point patches over a context of 4096 points, and predicts the next patch from the ones before it 2505.14766 §3.1 2505.14766 app. A.6. Each series is scaled with a mean and a standard deviation from the past and the current patch only, so that scaling does not leak future values 2505.14766 §3.1 2505.14766 app. A.1. Eleven time-wise causal layers alternate with one variate-wise layer, which uses a block-diagonal mask so that packed series do not attend to each other 2505.14766 §3.1 2505.14766 app. A.2. Each output is a mixture of 24 Student-T components 2505.14766 app. A.6.

Author-reported, on the BOOM benchmark Toto has the lowest MASE (0.617) and CRPS (0.375) 2505.14766 tab. 2. On GIFT-Eval it has the best average rank, while its CRPS is second to one baseline 2505.14766 tab. 3. In the ablation, held-out NLL rises by 27.3% without causal scaling and by 27.2% with one Student-T in place of the mixture 2505.14766 app. E. BOOM uses Datadog's staging data, and the company's own telemetry makes up 43% of the pretraining mix, so neither can be rebuilt from the public release 2505.14766 §4.1 2505.14766 §3.2. The rotary and XPOS positions are covered on the relative positional encoding page.

Frozen language model reprogrammed for forecasting (Time-LLM)

Time-LLM: Time Series Forecasting by Reprogramming Large Language Models keeps a Llama-7B backbone frozen and trains only an input layer and an output projection 2310.01728 §3. Each channel is normalised, cut into patches, and cross-attended onto learned text prototypes. The paper uses 1000 prototypes for most long-term tasks and 100 for ILI and short-term tasks [arxiv:2310.01728 §3.1] 2310.01728 app. B.4. A text prompt with dataset context, a task instruction and input statistics is prepended. The patch outputs are then flattened and projected to the horizon 2310.01728 §3.1 [arxiv:2310.01728 app. B.1].

Author-reported, with input length 512, the paper counts Time-LLM as best on seven of eight long-term datasets. ETTh2 is the exception, where PatchTST has the lower MSE, and Weather is a tie 2310.01728 §4.1. Few-shot results, on the first 10% of training steps, are about 5% lower MSE than GPT4TS 2310.01728 §4.3. Zero-shot transfer between ETT datasets is reported at 22% lower overall error than GPT4TS 2310.01728 §4.4. The model analysis gives 9.2% higher average MSE without patch reprogramming, and 10.2% higher without the statistics in the prompt 2310.01728 §4.5. Training memory is dominated by the backbone: 32,136 MiB at ETTh1-96 with 32 layers, against 6.39 M trainable parameters in the reprogramming layers [arxiv:2310.01728 §4.5]. Baseline numbers are partly copied from the GPT4TS paper and were not re-run 2310.01728 §4. The released code departs from the paper in several places (see Contradictions) 2310.01728 Paper vs code. The card is Time-LLM: Time Series Forecasting by Reprogramming Large Language Models. For the general idea of reusing pretrained models, see the foundation model and transfer learning pages.

Probabilistic diffusion per time step (TimeGrad)

TimeGrad 2101.12072 §3 factors the future of a multivariate series into one conditional distribution per time step. An LSTM summarises the past, and a conditional denoising diffusion model samples each next value from that summary 2101.12072 §3 2101.12072 eq. 9. Training uses the usual noise-prediction loss on random diffusion steps 2101.12072 §3.1 2101.12072 alg. 1. The paper's setup has two LSTM layers of width 40, Adam at learning rate 1e-3 and batch size 64 2101.12072 §4.2.

At forecast time each sample path runs 100 reverse diffusion steps per forecast step, and 100 paths are drawn 2101.12072 §4.2 2101.12072 alg. 2. Author-reported, the error measure CRPS_sum is lowest for TimeGrad on Solar, Electricity, Traffic, Taxi and Wikipedia. On Exchange, the smallest set, VES, VAR and a Transformer with masked autoregressive flows reach 0.005, and TimeGrad reaches 0.006 2101.12072 tab. 2. Sampling is the cost: each forecast step calls the denoiser once per diffusion step, so generating a horizon of P steps takes about P × N denoiser calls per sample batch 2101.12072 §6. The card is Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting. Background on the sampler is on the diffusion models page.

Diffusion over future positions (MID)

Stochastic Trajectory Prediction via Motion Indeterminacy Diffusion (MID) treats the future path as the output of a reverse diffusion. A fixed forward chain adds Gaussian noise over K = 100 steps 2203.13777 §3.2 2203.13777 eq. 2. A learned reverse chain, conditioned on an encoding of the observed path and of neighbours, removes the noise 2203.13777 §3.2 2203.13777 §3.5. Training uses the mean squared error between the true and predicted noise 2203.13777 eq. 10. The denoiser is a three-layer Transformer encoder of width 512 2203.13777 §3.5 2203.13777 app. B.

Sampling calls the denoiser once per step, so 100 calls per sample. Best-of-20 draws 20 samples per history 2203.13777 §4.1 2203.13777 alg. 2. Author-reported: on the Stanford Drone Dataset it reaches ADE and FDE of 7.61 and 14.30 pixels, the best trajectory-only result in its table 2203.13777 tab. 1. On ETH/UCY the average is 0.21 and 0.38 metres. The paper calls this comparable rather than best, since Y-Net with TTST uses images and 10,000 samples 2203.13777 §4.2 2203.13777 tab. 2. Removing the diffusion model does not recover the gain 2203.13777 §4.3 2203.13777 tab. 3. For 512 trajectories on ZARA1, the paper reports 0.443 s for Trajectron++ and 17.368 s for MID 2203.13777 §5.

The paper diffuses future positions 2203.13777 §3.1 2203.13777 eq. 1. The released code diffuses velocities instead, and its numbers depend on that choice (see the card's Paper vs code).

Trajectory data and a baseline (Lyft L5)

One Thousand and One Hours: Self-driving Motion Prediction Dataset releases about 1,118 hours of perception output from 20 vehicles on one route 2006.14480 §3.1 2006.14480 tab. 2. Its forecasting baseline is a ResNet-50 on a 224×224 bird's-eye-view raster centred on each agent. It predicts (x, y) over 5 seconds, with an L2 loss, 38,000 iterations and batch size 64 2006.14480 §4 2006.14480 fig. 7.

Author-reported, open-loop: displacement error (ADE) falls as training data grows. It is 2.47 m with 1% of the data and no history, and 0.77 m with the full data and 1 s of history 2006.14480 tab. 3. Planning is closed-loop and reported only as a qualitative run 2006.14480 fig. 8 2006.14480 §4. The dataset covers one route in daytime hours, so generalisation is untested 2006.14480 §3.1. The card does not establish a licence for the dataset, map or imagery, so check the release terms before reuse.

Entity-aware LSTM for many basins (EA-LSTM)

Towards Learning Universal, Regional, and Local Hydrological Behaviors via Machine-Learning Applied to Large-Sample Datasets trains one network on daily weather for 531 river basins, with 27 static catchment attributes 1907.08456 §2.4. In the EA-LSTM the attributes feed only the input gate, which is computed once per basin. The weather drives the other gates 1907.08456 eq. 7 1907.08456 §2.2. The loss, NSE*, scales each basin's squared errors by its discharge standard deviation, with a constant of 0.1 1907.08456 eq. 13 1907.08456 §2.3. Each sample is a 270-day window 1907.08456 §2.6.

Author-reported, on 447 basins, the mean validation NSE is 0.674 for one model and 0.705 for an eight-model ensemble. The LSTM with static inputs gets 0.685 for one model 1907.08456 tab. 3. The benchmark rows come from earlier studies run by other groups 1907.08456 §2.5. The paper argues that the small cost is worth the interpretability 1907.08456 §3.1. Both the raw-attribute clusters and the embedding clusters match geographic regions of the US, and the silhouette score peaks at k = 5 for the embedding 1907.08456 §3.4. Robustness is weaker: when the standardised attributes get noise of σ = 1, the median NSE falls from 0.71 to 0.48 1907.08456 §3.3.

The split runs backwards in time: trained on 1999–2008 and validated on 1989–1999 1907.08456 §2.6. The paper gives no learning rate, batch size or epoch count, and says stochastic gradient descent, while the code uses Adam and 30 epochs 1907.08456 Paper vs code. Hyperparameters were chosen by cross-validation on calibration data, not on the validation period 1907.08456 app. B.

Graph convolution with a learned graph (GCNN-DDGF)

The model forecasts the next hour of demand at all N stations at once, from the last 24 hourly vectors 1712.04997 §5 1712.04997 tab. 2. Its graph filter is not predefined. It is an N × N matrix of trainable parameters, learned together with the weights, so no station-link map is needed 1712.04997 §3.3.1 1712.04997 eq. 15. The four predefined graphs, built from distance, demand, trip duration or demand correlation, serve as baselines 1712.04997 §3.2.

The regular variant stacks graph-filter and feedforward blocks. The recurrent variant (GCNN_rec-DDGF) applies the filter at each hour and feeds an LSTM with 100 units 1712.04997 §3.3.2 1712.04997 tab. 2. On 272 Citi Bike stations, the recurrent variant has the lowest test RMSE of 13 models, 2.12, against 2.35 for the regular variant and 3.44 for the historical average. The four predefined-graph baselines fall between 2.50 and 3.44 [arxiv:1712.04997 tab. 3]. This is one split, author-reported, with no variance across runs 1712.04997 §5. The card is Predicting Station-level Hourly Demands in a Large-scale Bike-sharing Network.

Attention U-Net on radar maps (SmaAt-UNet)

The input is 12 five-minute radar maps stacked as channels on a 288 × 288 grid. The output is the rain map 30 minutes after the last input, pixel by pixel 2007.04417 §4.1. The network is a U-Net with CBAM attention on the skip connections and depthwise-separable convolutions in place of the regular ones 2007.04417 §3.1 2007.04417 §3.2. Because rain is rare, training keeps only samples whose target has at least half rainy pixels 2007.04417 §4.1.

Author-reported, on the 2019 NL-50 test set: its MSE of 0.0122 ties U-Net, and persistence scores 0.0248 2007.04417 tab. 4. It has 4.1 million parameters against 17.3 million for U-Net 2007.04417 tab. 1. The radar data are available only on request from the authors 2007.04417 §4.1. The card is SmaAt-UNet: Precipitation Nowcasting using a Small Attention-UNet Architecture. The attention and separable-convolution blocks are described on the attention mechanism and depthwise-separable convolution pages.

Learned global weather model on a mesh (GraphCast)

GraphCast: Learning skillful medium-range global weather forecasting is a one-step learned weather model on a 0.25° global grid, with six-hour steps 2212.12794 app. §2.2. It encodes the grid onto a multi-scale icosahedral mesh, runs 16 message-passing layers with unshared weights there, and decodes back to the grid 2212.12794 app. §3.4 2212.12794 app. §3.5 2212.12794 app. §3.6. The output is a scaled change added to the input state 2212.12794 app. §3.7. Forecasts run to 10 days by feeding each prediction back in 2212.12794 app. §3.1. Training uses a weighted mean squared error over rollouts of up to 12 steps, on ERA5 2212.12794 app. §4.2 2212.12794 app. §4.5.

Author-reported: GraphCast has lower RMSE than the operational HRES-fc0 on 90.3% of 1,380 targets over 10 days 2212.12794 §Forecast verification results 2212.12794 fig. 2. Optimal blurring of HRES narrows the gap, to 88.0% of targets, but does not close it 2212.12794 app. §7.4. A forecast takes under 60 seconds on one TPU v4 2212.12794 app. §3.2. Training took about four weeks on 32 TPU v4 devices 2212.12794 app. §4.7. The public repository has inference and loss code but no training loop (see Contradictions).

Traffic grid and graph benchmark (DL-Traff)

DL-Traff: Survey and Benchmark of Deep Learning Models for Urban Traffic Prediction adds no model. It surveys grid-based and graph-based traffic models, then runs them under one protocol on six datasets 2108.09091 §1 [arxiv:2108.09091 tab. 2]. Grid models predict the next inflow and outflow map from 6 past steps. Graph models predict 12 future speeds from 12 past speeds 2108.09091 §2 2108.09091 §5.1. The graph protocol uses a 7 : 1 : 2 split, Adam with learning rate 0.001, batch 64 and MAE loss by default, and early stopping after 10 epochs without validation gain, capped at 200 epochs 2108.09091 §5.1.

Author-reported: STDN has the lowest single-step grid RMSE on all four grid datasets 2108.09091 tab. 4. On the graph datasets, Graph WaveNet has the lowest 60-minute RMSE on METR-LA and MTGNN on PeMS-BAY [arxiv:2108.09091 tab. 5]. The authors do not call any grid model dominant 2108.09091 §5.2. The released grid code does not reproduce every reported setting, and it reports only MSE (see Contradictions).

Neural ODE over two road graphs (STGODE)

Spatial-Temporal Graph ODE Networks for Traffic Flow Forecasting predicts the next 60 minutes of flow at every sensor from the previous 60 minutes 2106.12931 §5.3. Graph propagation runs on two matrices: one from road distance, and one from DTW similarity of the traffic series 2106.12931 §4.1. Stacked graph layers are written as a tensor neural ODE with a restart term that keeps each sensor's input in play, so that deep stacks do not blur the sensors into one average 2106.12931 §4.2. Dilated temporal convolutions wrap each block 2106.12931 §4.3.

Author-reported, the table gives the lowest RMSE, MAE and MAPE on all six PeMS datasets 2106.12931 tab. 2. Some margins are small: PeMS04 RMSE is 32.82, against 33.65 for STSGCN 2106.12931 tab. 2. No variance, repeated runs or significance tests are reported 2106.12931 §5.4. The released code departs from the paper in ways that change the model. It adds a learnable scaling factor to the ODE, and its DTW graph uses a daily profile and a different threshold 2106.12931 Paper vs code.

Pretrained patch Transformer with a learned sparse graph (STEP)

Pre-training Enhanced Spatial-temporal Graph Neural Network for Multivariate Time Series Forecasting has two stages. First, a Transformer (TSFormer) is pretrained on a long history by reconstructing masked 12-step patches, with 75% of the patches hidden 2206.09113 §3.1. Then a Graph WaveNet forecaster reads the patch states of the last short window, together with a sparse graph learned from the series 2206.09113 §3.2. The pretraining is done separately for each dataset 2206.09113 §4.1.

Author-reported, STEP has the best MAE, RMSE and MAPE in nearly all cells of its table on METR-LA, PEMS-BAY and PEMS04, against 12 baselines [arxiv:2206.09113 §4.2] 2206.09113 tab. 2. The released code differs from the paper in ways that change the graph. It removes a dynamic term from the graph logits, uses one linear layer for the decoder, and keeps a global budget of 10N kNN edges rather than 10 per node 2206.09113 Paper vs code.

Fourier-domain graph layers over all window values (FourierGNN)

FourierGNN: Rethinking Multivariate Time Series Forecasting from a Pure Graph Perspective turns a window of N series with T steps into one fully connected graph of N·T nodes 2311.06190 §4.1. Each stacked layer multiplies the spectrum by a learned complex d-by-d matrix that all nodes share, so the cost grows as n log n rather than n² 2311.06190 §4.2 2311.06190 eq. 6.

Author-reported, at horizon 12 it has the lowest MAE and RMSE on all six datasets of its main table, and the lowest MAPE on four of them [arxiv:2311.06190 tab. 1] 2311.06190 §5.2. The released code differs from the equations. It uses diagonal weights, builds no adjacency matrix, scales only four of its seven datasets, and moves the model to cuda:0, so it will not run without CUDA 2311.06190 Paper vs code.

Multi-scale period graphs with attention (MSGNet)

MSGNet finds the k strongest periods of the input with an FFT and treats each period as one scale 2401.00423 §4.2. Each scale gets its own learned adjacency matrix and a Mixhop graph convolution, then self-attention within the period. The scales are summed with softmax weights from their amplitudes 2401.00423 §4.3 2401.00423 §4.4. The authors report the best average rank over eight datasets, 1.813 against 2.750 for TimesNet 2401.00423 tab. 1. Among the ablated variants, removing the learned graph gives the largest MSE on each of the three datasets in their ablation 2401.00423 tab. 2. The released code chains the scale graphs, applies a causal mask within each period and re-estimates the scales in every block. The paper describes none of these 2401.00423 Paper vs code.

Identity-embedding MLP without a graph (STID)

Spatial-Temporal Identity: A Simple yet Effective Baseline for Multivariate Time Series Forecasting forecasts many related series with plain MLP layers and no graph [arxiv:2208.05233 §3]. Each series gets a learned spatial identity vector, and each time-of-day and day-of-week slot gets one too. These are concatenated with a linear embedding of the history. Residual MLP blocks and a linear head then predict F = 12 future steps, trained with MAE 2208.05233 §3 2208.05233 eq. 5. The paper's hidden size is D = 32 2208.05233 §4.1.

Author-reported, from single runs with no variance given, STID has the lowest average MAE of the methods in its table on all five benchmarks (PEMS04, PEMS07, PEMS08, PEMS-BAY and Electricity) 2208.05233 tab. 3. It also has the lowest seconds per epoch on each of them 2208.05233 tab. 2. The paper's explanation is indistinguishable samples, and it gives no measure of that 2208.05233 §1. The released code adds dropout and weight decay, which the paper does not state, and its Electricity setup differs (see Contradictions) 2208.05233 Paper vs code.

Software library with a TCN example (Darts)

Darts: User-Friendly Modern Machine Learning for Time Series is a Python library, not a new model. It gives classical, deep and probabilistic forecasters one fit and predict interface. Each series is stored with its time axis, its components and any Monte Carlo samples in one container 2110.03224 §2.1 2110.03224 §2.2. Its one worked example trains a temporal convolutional network with a Laplace likelihood on two series and forecasts 48 steps with 500 samples 2110.03224 §3. The paper reports no accuracy benchmarks. The code is Apache-2.0 at a much later commit, so its model list and defaults differ from the paper's 2110.03224 Paper vs code.

Benchmark pipeline (BasicTS+), not a model

Exploring Progress in Multivariate Time Series Forecasting: Comprehensive Benchmarking and Heterogeneity Analysis adds no forecaster. It runs its 45-plus models through one pipeline: z-score scaling fitted on the training split, masked MAE for training, time-of-day and day-of-week features, and metrics computed on re-normalised data 2310.06119 §4.1 2310.06119 §4.2. The re-normalisation matters: for ETTh1, Autoformer's MAE is 0.483 on normalised values and 1.74 on the original scale 2310.06119 §4.2.

The paper sorts datasets by temporal pattern (clear, drifting or unclear) and by spatial indistinguishability, which it measures with cosine similarity between series 2310.06119 §5.1 2310.06119 §5.2. It tests four hypotheses. Advanced networks should help where patterns are clear, basic networks where they are not, and spatial modelling should help only where samples are indistinguishable 2310.06119 §5. In its tables, advanced models (Informer, Autoformer, FEDformer) beat the linear models on PEMS04 and PEMS08, and the linear models beat them on ETTh2 and ETTm2 2310.06119 §6.2. Removing the spatial component from STID and AGCRN makes them worse on METR-LA and PEMS-BAY, and better on ExchangeRate and ETTm1 2310.06119 §6.3. Every number in these tables comes from the authors' own runs through their pipeline.

Transformer on irregular triplets with forecasting pretext (STraTS)

Self-Supervised Transformer for Sparse and Irregularly Sampled Multivariate Clinical Time-Series (STraTS) takes each stay as a set of (time, variable, value) triplets, with no aggregation or imputation 2107.14293 §3.1. Each triplet is embedded from a learned variable lookup plus two continuous-value embeddings, one for time and one for value 2107.14293 §3.2.1. Then come Transformer blocks, attention pooling and a sigmoid head for in-hospital mortality 2107.14293 §3.2.2 2107.14293 §3.2.5 2107.14293 tab. 3.

Forecasting is the pretext task. The model predicts the variables observed in a later two-hour window, with a squared error over those variables. The whole model is then fine-tuned on the mortality label 2107.14293 §3.2.6. The paper chose forecasting over masking as the pretext 2107.14293 §3.2.6.

Author-reported: on MIMIC-III (24-hour mortality), ROC-AUC 0.891 and PR-AUC 0.577, against 0.886 and 0.559 for the best baseline, GRU [arxiv:2107.14293 tab. 4]. On PhysioNet-2012 (48-hour mortality), PR-AUC is 0.498, against 0.481 for GRU-D 2107.14293 tab. 4. The paper reports PR-AUC gains of 3.2% and 3.5% and says STraTS is best on all metrics on both sets 2107.14293 §4.5. Each model is trained 10 times on a random 50% of the labelled data 2107.14293 §4.5. Without self-supervision, MIMIC-III PR-AUC falls to 0.546 2107.14293 tab. 5. The label-fraction curves are in figures only 2107.14293 fig. 4 2107.14293 fig. 5.

Large models for time series (survey)

Large Models for Time Series and Spatio-Temporal Data: A Survey and Outlook has no experiments and proposes no model. It splits the field into time series and spatio-temporal data, and within each, into language-model and foundation-model approaches 2310.10196 §3. Use it as a reading list and a vocabulary. Its coverage claims are the authors' own 2310.10196 tab. 1.

Graph surveys for time series (GNN4TS and urban STGNNs)

Four surveys map graph-based models. None runs experiments, so they give no new numbers. A Survey on Graph Neural Networks for Time Series: Forecasting, Classification, Imputation, and Anomaly Detection covers forecasting, classification, anomaly detection and imputation. It gives a task taxonomy and an architecture taxonomy 2307.03759 §3. Its graph rules are inverse distance, a 0/1 adjacency, series similarity and Granger causality, plus learned graphs 2307.03759 §2. Spatio-Temporal Graph Neural Networks for Predictive Learning in Urban Computing: A Survey covers city data, mainly traffic. It sorts graph construction into topology, distance, similarity and interaction rules 2303.14483 §2. Its framework has a data-processing module, a spatio-temporal graph learning module and a task-aware prediction module 2303.14483 §5. Transportation is over 60% of the surveyed applications 2303.14483 §4. Graph Neural Network for Traffic Forecasting: A Survey covers 212 papers from 2018 to 2020 on traffic forecasting. It sorts traffic graphs into road, region and station levels, and adjacency matrices into road-based, distance-based, similarity-based and dynamic types 2101.11174 §4.1.1 2101.11174 §4.1.2. Its benchmark table is copied from the surveyed papers, which the authors say are not directly comparable [arxiv:2101.11174 §5.3] 2101.11174 tab. 11.

A Review of Graph Neural Networks and Their Applications in Power Systems covers six families of GNN, including GCN, GAT and spatio-temporal GNNs, and surveys their use in power systems: fault diagnosis, time-series forecasting, power flow and data generation 2101.10025 §II. It runs no experiments of its own, and each number it quotes comes from another paper and setup 2101.10025 Method. Use it for power-grid graphs, which the traffic surveys above do not cover.

Forecast residuals for anomaly detection (GDN)

Graph Neural Network-Based Anomaly Detection in Multivariate Time Series learns the sensor graph as well as the forecaster. Each sensor has a learned embedding, and its neighbours are the top k sensors by cosine similarity of embeddings 2106.06947 §3.3 2106.06947 §3.4. A graph-attention forecaster predicts each sensor's next value from its own window and its neighbours' 2106.06947 §3.5. A time tick scores the largest sensor error, normalised by that sensor's median and IQR, then smoothed. The tick is anomalous when the score passes a threshold 2106.06947 §3.6. Training uses normal data only 2106.06947 §3.1.

Author-reported, the test-set F1 is 0.81 on SWaT and 0.57 on WADI, with precision 99.35 and 97.50 2106.06947 tab. 2. Replacing the top-k graph with a complete graph lowers F1 to 0.78 and 0.51 2106.06947 tab. 3. Each result comes from one configuration with no standard deviation 2106.06947 tab. 2. SWaT and WADI are not in the repository, which ships only a demo subset 2106.06947 Pitfalls.

Forecast of a simulated resilience curve (LSTM)

A Resilience Recovery Method for Complex Traffic Network Security Based on Trend Forecasting models failure spread with an SIRD-type model that gives each node a risk value 2609.27903 §3.1. It computes a resilience curve E(t) from network performance 2609.27903 §3.2. An LSTM forecasts that curve, and the forecast triggers protection or recovery of nodes 2609.27903 §3.3 2609.27903 §3.4. On one simulated traffic network, the LSTM reaches R2 0.96842 and MSE 0.00016. An RNN reaches 0.91152 and 0.00051, and a CNN 0.85833 and 0.00082 2609.27903 tab. 4. With forecast time advance 0.3, overall resilience rises from 0.7261 to 0.9354 2609.27903 tab. 8.

The paper does not state the LSTM's window, split, layers, optimiser or epochs 2609.27903 §3.3. The forecast curves overlap the data closely, so the errors may be in-sample 2609.27903 fig. 12. The CNN and RNN baselines have no stated architecture 2609.27903 tab. 4. There is no official code, and every table entry is a single run 2609.27903 Paper vs code. Treat it as a worked pipeline, not a forecasting benchmark.

When to use which

The paper's own conclusion is that the choice depends on the required accuracy, the speed of forecasts and the tuning budget 2103.12057 §4.3. The rows below follow the cards' results. They are not a verdict for other data, which the cards do not test.

Whichever you pick for the benchmark, start from the benchmark settings that were best for each architecture: a past-history factor of 1.25 × H, learning rate 0.001, and batch size 32 or 64 depending on the model 2103.12057 tab. 18. Then check the choice on your own series, because the paper's best cells were selected on the test split.

Implementation sketch

Language-agnostic outline of the benchmark protocol for the scalar-series variants. The steps are shared by the benchmark architectures; only the forecaster in step 4 differs. The HTM, Transformer, trajectory, graph, grid and many-series variants use different inputs, targets and losses, so follow their cards. TimeMixer and TabPFN-TS also change the steps: TimeMixer needs per-scale normalisation, and TabPFN-TS builds time features instead of raw windows, and WSB adds a booster after each recursive step. Time-LLM replaces the forecaster with a frozen LLM that reads reprogrammed patches 2310.01728 §3.1. LTSF-Linear reads a fixed 336 steps by default, and NLinear subtracts the last observed value before its layer and adds it back after 2205.13504 app. B.2 2205.13504 §4. DCGRU adds class features to each row, TiDE takes covariates (and a series index in the released code), and Toto scales each patch from the past only. STID adds learned identity tables to the embedded history, TSMixer normalises each window and mixes it across time and variates, and FreTS works on the Fourier transform of the input.

# Scale each series with statistics of its own training part  [§3.3]
def normalize(train_part, method):
    # Min-max or z-score; use training statistics only  [§3.3]
    ...

# Build (window, horizon) training pairs by sliding one step  [§3.3] [fig. 2]
def make_windows(series, past_history, horizon):
    # past_history = 1.25 * horizon in the best settings  [tab. 18]
    ...

# Map a window of shape (L, 1) to H outputs  [§3.2.1]
def forecaster(window, horizon):
    # Any variant: MLP, recurrent cell, CNN or TCN, then dense(H)
    ...

# Train with Adam; the paper uses five epochs  [§3.3] [tab. 10]
def train(model, windows, targets, batch_size, learning_rate):
    # Learning rate 0.001 and batch 32 or 64 were the best  [tab. 18]
    ...

# Score one forecast origin per series with WAPE  [§3.3.1] [eq. 4]
def wape(actual, forecast):
    # Mean absolute error divided by the mean of the actual values
    ...

The paper does not name the training loss 2103.12057 §3.3. Choose it yourself and record the choice.

Contradictions

Several cards conflict with themselves, with each other, or with their released code. Both sides are shown; this page does not resolve them.

• Layer count. The text says fewer stacked recurrent layers did better, except for ESN 2103.12057 §4.4.2. The LSTM row of the selection table marks two layers as significantly best and one layer as a tendency 2103.12057 tab. 17.

• TCN return sequence. The text says returning the full sequence hurts the TCN 2103.12057 §4.4.3 2103.12057 §4.5. The selection table lists the return-sequence setting as true for the TCN [arxiv:2103.12057 tab. 17]. The best TCN in its own table returns the last output only 2103.12057 tab. 15.

• Grid size. The paper gives 6,432 configurations per dataset, and also more than 38,000 models in total over the datasets 2103.12057 §1 2103.12057 §3.2. Counting the grids in its tables gives about 3,216 per dataset (my count) 2103.12057 tab. 9 2103.12057 tab. 10. The paper does not explain the factor of two.

• Paper versus code. The released code does not match the paper in several places. The z-score divides by the standard deviation, while the paper's equation divides by the range 2103.12057 eq. 3. The shipped CNN grid stops at three layers, so the paper's four-layer best model is not in that grid 2103.12057 tab. 9. The shipped training grid is also smaller than the paper's. See the card's Paper vs code section before reproducing the numbers.

• Recurrent graph model. The bike paper reports the recurrent variant as best 1712.04997 tab. 3. Its public repository, as reviewed on the card, holds only the regular variant for this data. The recurrent result therefore cannot be checked against the released code. The card also lists several training settings where the code differs from the paper (see its Paper vs code section).

• Linear models versus Transformers on long-term forecasting. The BasicTS+ paper says earlier reports found linear models ahead of Transformers, and later reports found the reverse 2310.06119 §1. It attributes the conflict to dataset heterogeneity, and does not settle it 2310.06119 §5. Its own table shows advanced models ahead on PEMS04 and PEMS08, and linear models ahead on ETTh2 and ETTm2 2310.06119 §6.2. LTSF-Linear reports the other direction on its own nine benchmarks: its best linear model beats the best Transformer baseline in all 36 cells of its table 2205.13504 tab. 2. Those Transformer baselines use a look-back of 96 against 336 for the linear models, and mostly come from another paper 2205.13504 app. B.2 2205.13504 tab. 2. The two papers test different protocols and datasets, so their results are not a single verdict.

• Released GraphCast weights. The paper describes models trained on ERA5 only. The released demo lists a model fine-tuned on HRES data from 2016 to 2021, which the paper does not describe 2212.12794 Paper vs code. Its scores are not the paper's scores.

• Text against table in BasicTS+. The text says PatchTST and STID trail classic methods on M4. The sMAPE row puts both ahead of LGBM and DeepAR, and the claim holds only against NBeats 2310.06119 §6.4 (M4 table). The card's Pitfalls section lists this and a similar D2STGNN mismatch.

• Traffic training settings. The paper gives learning rate 0.001, batch 64 and MAE loss as defaults for all models 2108.09091 §5.1. The released grid code uses learning rate 0.0001, batch 1024 and MSE loss for STDN, and reports only MSE. The paper's grid MAE and MAPE therefore cannot be checked against that code 2108.09091 tab. 4.

• STDN flow gating. The paper describes STDN with flow gating 2108.09091 §4.1. The released grid code has only the no-flow STDN script, so the flow-gated numbers have no matching code 2108.09091 §4.1.

• Scaler and test leakage. The released graph code standardises each sensor with statistics from the whole series, test period included. The paper does not say which was used 2108.09091 §5.1.

• STraTS attention and residuals. The paper scales attention by the square root of d/h and adds residual connections with layer normalisation after each sub-layer 2107.14293 §3.2.2. The released code does not scale, averages the residual branch and drops layer norm, and its own README says results deviate from the published ones 2107.14293 tab. 4.

• Anomaly threshold in GDN. The paper sets the threshold to the largest smoothed validation score 2106.06947 §4.3. The released code's default instead picks the threshold that maximises F1 on the test labels 2106.06947 Paper vs code. The paper does not say which mode produced its table, so the reported F1 may be optimistic.

• EA-LSTM against its own LSTM. The main claim is that EA-LSTM beats the benchmark hydrological models 1907.08456 §3.2. In the paper's own table, the EA-LSTM single model has a lower mean NSE than the LSTM with static inputs, 0.674 against 0.685 1907.08456 tab. 3.

• Resilience text against its table. The text calls R2 close to 1 under all six conditions 2609.27903 §4.2. Table 5 gives 0.87052 to 0.97014 2609.27903 tab. 5.

• Resilience recovery term. The printed equation 3 gives dR/dt = μ0 · I, but dI/dt subtracts only (1 − f) μ0 · I, so the total node count is not conserved. Figure 4 and the text use (1 − f) μ0 for the recovery flow 2609.27903 eq. 3 2609.27903 fig. 4.

• TiDE residual blocks. The paper describes multiple residual blocks and tunes one to three layers 2304.08424 §4.1 2304.08424 tab. 7. The released code builds one fewer block than the layer setting. Its published Traffic, Weather and ETTm1 settings therefore have no residual block in the encoder or decoder 2304.08424 Paper vs code 2304.08424 tab. 8.

• TiDE look-back on ETTh1. The paper says the look-back is 720 for every horizon 2304.08424 §5.1. The ETTh1 script sets 720 and then 336 for the same command, which in absl flags would keep 336 at horizons 336 and 720. The card did not run the code to confirm this 2304.08424 Paper vs code.

• Toto ranks. The BOOM table gives mean rank 2.369 for Toto, and the text gives 2.351 2505.14766 tab. 2 2505.14766 §5. The repository leaderboard lists the same MASE and CRPS with a rank of 6.939 2505.14766 Paper vs code.

• Toto causality. The paper says statistics come from the history only at inference 2505.14766 §3.1. In the released training path, the clamp uses the whole input window, so training sees future points 2505.14766 Paper vs code. The paper calls this relaxed causality.

• STGODE sensor count. The text gives 288 sensors for PeMSD7(M), and the table and code give 228 2106.12931 §5.1 2106.12931 tab. 1.

• STGODE temporal convolutions. The paper describes a residual dilated stack for every TCN 2106.12931 §4.3. In the released code, a residual shortcut skips the dilated stack for most TCN instances: 6 of 24 run it 2106.12931 Paper vs code.

• STGODE semantic threshold. The paper links pairs whose DTW distance is below 0.6 2106.12931 eq. 6. The released code applies a Gaussian kernel first, which by the card's own derivation links only pairs within about 0.07 standard deviations of the mean. This derivation was not run 2106.12931 Paper vs code.

• DCGRU text against its table. Section 6 says the Scenario 1 MAE of DCGRU and EDCGRU does not exceed 1, but table 6 gives 1.11 for DCGRU on area 2 2512.06714 §6 2512.06714 tab. 6.

• DCGRU error reduction. The conclusion says virtual readings cut error by about 30%. The table gives drops of 22% to 31%, depending on the metric and area, and the paper does not say which one the 30% refers to 2512.06714 §7 2512.06714 tab. 6.

• DCGRU dense activations. Section 4.1 says the second dense layer is linear, and table 5 lists ReLU 2512.06714 §4.1 2512.06714 tab. 5.

• STID Electricity setup. The paper gives P = 168 and F = 12 for Electricity 2208.05233 §4.1. The released config uses 336 and 336 2208.05233 Paper vs code, and the paper does not say which setup produced its Electricity column.

• STID input. The paper embeds only the traffic history [arxiv:2208.05233 §3] 2208.05233 eq. 1. The code also flattens the time-of-day and day-of-week channels into the embedding 2208.05233 Paper vs code.

• TSMixer split. The paper states 7:2:1 for Weather, Electricity and Traffic 2303.06053 tab. 2. The release gives 70% train, 10% validation and 20% test 2303.06053 Paper vs code.

• TSMixer normalisation. The paper's equation puts normalisation after the residual, while its text says pre-normalisation is used for long-term forecasting 2303.06053 app. B.3. The release is pre-norm and uses BatchNorm, not the LayerNorm the equation needs [arxiv:2303.06053 Paper vs code].

• FreTS claim against its table. The text says FreTS "outperforms all baselines on all datasets" 2311.06184 §4.1. Table 2 has PatchTST ahead at several horizons 2311.06184 tab. 2.

• FreTS channel learner. The paper says the channel learner is part of the default model 2311.06184 app. B.3. The released code never runs it, so the ablation in 2311.06184 tab. 3 cannot be reproduced from it 2311.06184 Paper vs code.

• Time-LLM prototypes and projection. The paper uses 100 prototypes for short-term tasks 2310.01728 app. B.4. The code fixes 1000 rows for every task 2310.01728 Paper vs code. The paper flattens the full patch outputs before projecting 2310.01728 app. B.1, while the code keeps only part of each output 2310.01728 Paper vs code.

• Time-LLM training settings. The paper gives one initial learning rate and no schedule 2310.01728 app. B.4. The ETTh1 script uses 100 epochs, and its learning rate changes with the horizon 2310.01728 Paper vs code. The code's default backbone depth is 6 layers, while the scripts use 32 2310.01728 Paper vs code.

• CoST trend experts. The paper sets the number of experts from the input length, which gives kernels up to 64 for a 201-step input. The scripts use kernels 1 to 128, which is 8 experts 2202.01575 §3.1 [arxiv:2202.01575 Paper vs code]. The paper's batch size is 256, and the scripts pass 128 2202.01575 §4.1 2202.01575 Paper vs code. The paper also computes the frequency loss before the inverse FFT, and the code applies it to the time-domain seasonal output 2202.01575 §3.2 [arxiv:2202.01575 Paper vs code]. The card lists further differences, such as the projection head.

• WSB baselines against the booster. The paper does not say that the booster loop feeds one shared history to its models, while each baseline uses its own. The tuned-LSTM comparison therefore runs two different recursive setups 2605.25305 Pitfalls. The released code also rounds single-model forecasts to integers, but not the booster's inputs 2605.25305 Paper vs code.

• Electricity text and tables. The paper gives the Coronel booster MAE as 464.93 in its table and 465.02 in the abstract. The saved output has 465.019 2605.25305 tab. 15 2605.25305 Paper vs code. The text gives the Palmas cross-validation MAE of GA-RF as 758.80, while its table gives 1052.60 2605.25305 §4.1 2605.25305 tab. 13. The GA-ESN row for Palmas and the ESN row for Coronel differ from the saved output 2605.25305 Paper vs code.

Across cards, the results do not measure the same thing, so they are not rankings against each other:

• LSTM on taxi counts versus LSTM on the benchmark. On NYC taxi counts, HTM is comparable to the best LSTM, with MAPE and negative log-likelihood 1512.05463 §5. On the benchmark, LSTM is best on 4 of 12 datasets by WAPE 2103.12057 §4.1. The datasets, buffers and metrics differ.

• Noise. The HTM card reports that LSTM is slightly better under temporal noise during training 1512.05463 §4.5. The benchmark does not test noise 2103.12057 §3.1.

• Speed. The benchmark ranks CNN as the most efficient deep model 2103.12057 §4.2. The diffusion card gives a cost for its own sampler, not for the benchmark models 2203.13777 §5. The times were measured on different hardware and are not comparable.

• Weather scores. GraphCast reports RMSE and anomaly correlation against ERA5 and HRES-based ground truth, over 1,380 targets [arxiv:2212.12794 app. §5.2]. These are not comparable with the MAE and WAPE figures on the traffic and energy benchmarks.

• Probabilistic and point scores. TimeGrad's CRPS_sum and the benchmark's point errors measure different things. A sampled distribution is not scored the same way as a single forecast 2101.12072 §4.1 2103.12057 §3.3.1.

• Forecast error versus mortality AUC. STraTS's forecasting head is only a pretext task. Its ROC-AUC and PR-AUC are classification scores and are not comparable with the forecast errors of the other cards 2107.14293 §3.2.6 2103.12057 §3.3.1.

• Task scores. NSE on basins, F1 on sensor anomalies and R2 on a resilience curve each score a forecast inside a larger task 1907.08456 tab. 3 2106.06947 tab. 2 2609.27903 tab. 3. None is comparable with the MSE, MAE, WAPE or CRPS_sum of the forecasting cards.

• LLM protocol. Time-LLM uses input length 512 2310.01728 §4.1. The linear models use 336 and the Transformer baselines 96 2205.13504 app. B.2. Its gains are measured against its own baselines, which are partly copied from another paper 2310.01728 §4. Its MSE is not a ranking against the linear-model or benchmark cards.

• Scaled scores. TiDE reports MSE on standardised targets [arxiv:2304.08424 §5.1], and Toto's MASE and CRPS use its own scaling and aggregation 2505.14766 app. C.2.4. Neither is comparable with the MAE and MAPE of the traffic and water cards, which use other protocols 2106.12931 tab. 2 2512.06714 tab. 6.

• Graph models against their code. STEP's paper gives an MLP decoder and a summed pairwise graph loss 2206.09113 §3.1 2206.09113 eq. 3. The released code uses one linear layer and a mean BCE loss [arxiv:2206.09113 Paper vs code]. FourierGNN's paper min-max normalises all seven datasets 2311.06190 §5.1, while its code scales four of them [arxiv:2311.06190 Paper vs code].

• Numbers from different lookback choices. ETSformer and MSGNet both report results on ETTm2 and ECL at horizon 96. ETSformer gives MSE 0.189 and 0.187, and MSGNet gives 0.177 and 0.165 2202.01381 tab. 1 2401.00423 tab. 1. ETSformer chooses its lookback per setting on validation MSE 2202.01381 app. C.1. MSGNet fixes the input at 96 2401.00423 §5.3. The protocols differ on this point, so the gaps do not show one model beating the other under a shared setup.

Papers

• An Experimental Review on Deep Learning Architectures for Time Series Forecasting: benchmarks seven deep architectures on 12 public univariate datasets with WAPE, using a global model per dataset and grid search.
• Predicting Station-level Hourly Demands in a Large-scale Bike-sharing Network: A Graph Convolutional Neural Network Approach: graph CNN whose graph filter is learned from data, with a recurrent variant, on 272 bike stations.
• SmaAt-UNet: Precipitation Nowcasting using a Small Attention-UNet Architecture: U-Net with CBAM attention and depthwise-separable convolutions for 30-minute radar rain maps.
• Continuous online sequence learning with an unsupervised neural network model: HTM sequence memory that learns one element at a time without gradients, compared with ARIMA, ELM, ESN and LSTM on taxi counts.
• Transformers in Time-series Analysis: A Tutorial: tutorial on Transformer building blocks and time-series variants, with no experiments of its own.
• Stochastic Trajectory Prediction via Motion Indeterminacy Diffusion: diffusion model over future pedestrian positions, evaluated with Best-of-20 on SDD and ETH/UCY.
• One Thousand and One Hours: Self-driving Motion Prediction Dataset: open driving dataset with a ResNet-50 rasterised baseline for open-loop forecasting.
• Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting: TimeGrad, an LSTM with a per-step diffusion sampler for multivariate probabilistic forecasts.
• GraphCast: Learning skillful medium-range global weather forecasting: graph network on a multi-scale mesh that rolls out a global weather forecast autoregressively.
• Exploring Progress in Multivariate Time Series Forecasting: Comprehensive Benchmarking and Heterogeneity Analysis: BasicTS+ pipeline and a dataset split by temporal pattern and spatial indistinguishability.
• Large Models for Time Series and Spatio-Temporal Data: A Survey and Outlook: survey and taxonomy of language-model and foundation-model approaches, with no experiments.
• Graph Neural Network for Traffic Forecasting: A Survey: survey of GNN traffic forecasters, sorting traffic graphs, adjacency types and GNN families, with no experiments of its own.
• CoST: Contrastive Learning of Disentangled Seasonal-Trend Representations for Time Series Forecasting: contrastive trend and seasonal encoder with a ridge regressor on frozen features.
• Self-Supervised Transformer for Sparse and Irregularly Sampled Multivariate Clinical Time-Series: Transformer over (time, variable, value) triplets, pretrained by forecasting the next two hours, for mortality prediction.
• DL-Traff: Survey and Benchmark of Deep Learning Models for Urban Traffic Prediction: survey and benchmark of grid and graph traffic models under one protocol.
• Are Transformers Effective for Time Series Forecasting?: one-layer linear forecasters (LTSF-Linear, DLinear, NLinear) compared against published Transformer forecasters on nine benchmarks.
• TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting: MLP-only forecaster that mixes seasonal and trend parts across pooled scales and sums one predictor per scale.
• Towards Learning Universal, Regional, and Local Hydrological Behaviors via Machine-Learning Applied to Large-Sample Datasets: EA-LSTM whose input gate reads static basin attributes, trained on 531 CAMELS basins.
• Graph Neural Network-Based Anomaly Detection in Multivariate Time Series: learned sensor graph and graph-attention forecaster, scored by forecast deviation for anomaly detection.
• From Tables to Time: Extending TabPFN-v2 to Time Series Forecasting: zero-shot forecasting by casting time steps as table rows for a pretrained tabular model.
• A Resilience Recovery Method for Complex Traffic Network Security Based on Trend Forecasting: SIRD-R failure model with an LSTM that forecasts a resilience curve to trigger node protection and recovery.
• Spatial-Temporal Graph ODE Networks for Traffic Flow Forecasting: neural ODE over a distance graph and a DTW graph, with dilated temporal convolutions, for six PeMS traffic datasets.
• Long-term Forecasting with TiDE: Time-series Dense Encoder: MLP encoder-decoder with covariates and a linear shortcut, for long-horizon forecasting.
• This Time is Different: An Observability Perspective on Time Series Foundation Models: decoder-only foundation model for observability metrics, with causal patch scaling and a Student-T mixture output, plus the BOOM benchmark.
• A Novel Deep Neural Network Architecture for Real-Time Water Demand Forecasting: GRU with k-means class features and a virtual-value variant, for 15-minute district water demand.
• Spatial-Temporal Identity: A Simple yet Effective Baseline for Multivariate Time Series Forecasting: graph-free MLP with learned spatial and temporal identity embeddings, evaluated on five traffic and energy benchmarks.
• TSMixer: An All-MLP Architecture for Time Series Forecasting: all-MLP forecaster that alternates time mixing and feature mixing, with a linear analysis and an M5 extension.
• A Survey on Graph Neural Networks for Time Series: Forecasting, Classification, Imputation, and Anomaly Detection: survey of GNNs for time series across four tasks, with task and architecture taxonomies.
• Spatio-Temporal Graph Neural Networks for Predictive Learning in Urban Computing: A Survey: survey of spatio-temporal GNNs for city data, mostly traffic, with graph-construction and module taxonomies.
• Frequency-domain MLPs are More Effective Learners in Time Series Forecasting: FreTS, two complex-valued MLPs on the Fourier transform of the input, for short- and long-term forecasting.
• Time-LLM: Time Series Forecasting by Reprogramming Large Language Models: frozen Llama-7B with learned text prototypes and a text prompt, for long-term, short-term and few-shot forecasting.
• A Review of Graph Neural Networks and Their Applications in Power Systems: review of six GNN families and their power-system uses, with no experiments of its own.
• Pre-training Enhanced Spatial-temporal Graph Neural Network for Multivariate Time Series Forecasting: STEP, a masked-patch pretrained Transformer feeding a learned sparse graph into Graph WaveNet, for traffic forecasting.
• FourierGNN: Rethinking Multivariate Time Series Forecasting from a Pure Graph Perspective: one fully connected space-time graph of the window, with graph layers in the Fourier domain.
• Electricity Consumption Forecasting: An Approach Using Cooperative Ensemble Learning with SHapley Additive exPlanations: GA-tuned LSTM, Random Forest, SVR and XGBoost with SHAP pruning, plus a booster from weaker forecasts, for monthly campus electricity.
• Darts: User-Friendly Modern Machine Learning for Time Series: Python library with one fit/predict API over classical, deep and probabilistic forecasters, with a TCN example and no accuracy benchmark.
• ETSformer: Exponential Smoothing Transformers for Time-series Forecasting: Transformer whose attention is exponential smoothing for trend and Fourier bases for season, with a level, growth and season decomposition.
• MSGNet: multi-scale inter-series correlations; learns one graph and self-attention per FFT-detected period scale and sums the scales by amplitude softmax.