---
kind: concept
slug: reduced-order-modeling
title: Reduced-order modeling
summary: >-
  Replaces a large, slow simulation with a small model that keeps its dominant behaviour, so design loops and many-run studies become affordable. Variants: intrusive POD-Galerkin, which projects the governing equation onto spatial modes; non-intrusive polynomial chaos fitted to snapshots; and data-driven reduced models. The only source so far is one 2021 review with no experiments.
tags: [reduced-order-model, surrogate, physics-informed, civil-engineering]
papers: ["2110.04600"]
related: [surrogate-modeling, dimensionality-reduction, dynamic-mode-decomposition, digital-twin, physics-informed-neural-network, gaussian-process, informed-machine-learning]
status: draft
updated_at: 2026-11-08
---

## Overview

A reduced-order model (ROM) replaces a high-fidelity simulation with a smaller
model that keeps the behaviour that matters [arxiv:2110.04600 §2]. The
motivation given is that full physics simulations are too slow to run inside an
iterative design loop, and that uncertainty quantification needs many runs
[arxiv:2110.04600 §1]. The shared idea is to find a low-dimensional basis from
simulation snapshots, then describe the system with a few coefficients
[arxiv:2110.04600 §3, eq. 9]. The variants below differ in how those
coefficients are obtained: by projecting the governing equation, by fitting a
surrogate to snapshots, or by learning the reduced dynamics from data.

## Variants

### Intrusive POD-Galerkin

Proper orthogonal decomposition (POD) takes modes from snapshots of a
high-fidelity solution, through an SVD or an eigendecomposition of the snapshot
matrix [arxiv:2110.04600 §3]. The field is then approximated as a truncated sum
of modes multiplied by time coefficients [arxiv:2110.04600 §3, eq. 9]. The
intrusive step projects the governing equation onto the modes and solves for
the coefficients [arxiv:2110.04600 §3.2, eq. 10-11]. The paper gives no discrete
algorithm and no rule for choosing the truncation length L [arxiv:2110.04600
§3]. Card: [A Review of Physics-based Machine Learning in Civil Engineering](/papers/2110.04600).

### Non-intrusive polynomial chaos

A non-intrusive method does not touch the governing equation. It fits a
surrogate from snapshots, here a polynomial chaos expansion (PCE) with Hermite
polynomials [arxiv:2110.04600 §3.3, eq. 12]. The index set of the expansion is
named but not defined, and the coefficient estimation method is not described
[arxiv:2110.04600 §3.3]. Card: [A Review of Physics-based Machine Learning in Civil Engineering](/papers/2110.04600).

### Data-driven reduced models

Data-driven ROMs learn either the mapping from the full state to the reduced
state, or the reduced model itself, from data [arxiv:2110.04600 §3.4]. The
source gives no architecture, training recipe or error figures for this
variant. Card: [A Review of Physics-based Machine Learning in Civil Engineering](/papers/2110.04600).

## When to use which

| Situation | Use | Why |
|---|---|---|
| You can change or re-derive the governing equations and the solver | Intrusive POD-Galerkin | The equation is projected onto the modes [arxiv:2110.04600 §3.2, eq. 10-11] |
| The solver is a black box and you only have its outputs | Non-intrusive PCE | Needs only snapshots; the surrogate is fitted from them [arxiv:2110.04600 §3.3] |
| You have many snapshots but no usable reduced equations | Data-driven ROM | Learns the reduced model or mapping from data [arxiv:2110.04600 §3.4] |
| You need a speed-up or accuracy figure to commit to a method | None of these from this source | The review reports no benchmark numbers [arxiv:2110.04600 tab. 1, fig. 6] |

The source calls POD-based ROMs "still state-of-the-art" [arxiv:2110.04600
§3], but gives no comparison that supports the claim. Treat it as the author's
opinion. The PCE branch has the most open points: the index set is undefined,
and the stability and curse-of-dimensionality issues are raised without
quantification [arxiv:2110.04600 §3.3]. Check the source before implementing
either branch.

## Implementation sketch

The two branches share the modal basis. Only the coefficient step differs.
The sketch is our own notation. Where the paper leaves a choice open, the
comment says so.

```pseudocode
# Stack high-fidelity snapshots as columns  [arxiv:2110.04600 §3]
S = [u(t_1), ..., u(t_M)]                  # N x M

# Modes from the SVD (or eigendecomposition of S^T S)  [arxiv:2110.04600 §3]
Phi, sigma = svd(S)

# Keep L modes. The paper does not give a choice of L  [arxiv:2110.04600 eq. 9]
Phi_L = Phi[:, :L]

# Approximate the field as a sum of modes times coefficients  [arxiv:2110.04600 eq. 9]
# u(x, t) ~ sum_k a_k(t) * phi_k(x)

# Intrusive branch: project the governing equation onto Phi_L and integrate
# the reduced ODE for a(t)  [arxiv:2110.04600 §3.2, eq. 10-11]
a = integrate(project(governing_eq, Phi_L), a_0)

# Non-intrusive branch: fit a surrogate from snapshot parameters to a_k
# The expansion (Hermite PCE) is given, but its fitting method is not  [arxiv:2110.04600 §3.3]
coeffs = fit_surrogate(snapshot_params, a_snapshots)
```

## Contradictions

None known across cards. The only source is a review with no experiments, so
it has no results that conflict with another paper. Inside that source, eq. 12
is used for both the PCE expansion and a Gaussian process mean, so cite by
section when reading it [arxiv:2110.04600 §3.3] [arxiv:2110.04600 §4.2].

## Papers

- [A Review of Physics-based Machine Learning in Civil Engineering](/papers/2110.04600): a 2021 review of ROMs (POD-Galerkin, PCE, data-driven), physics-informed GPs and PINNs in civil engineering. It runs no experiments and has known errors in its equations and citations.