Reduced-order modeling

Overview

A reduced-order model (ROM) replaces a high-fidelity simulation with a smaller model that keeps the behaviour that matters 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 2110.04600 §1. The shared idea is to find a low-dimensional basis from simulation snapshots, then describe the system with a few coefficients 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 2110.04600 §3. The field is then approximated as a truncated sum of modes multiplied by time coefficients 2110.04600 §3, eq. 9. The intrusive step projects the governing equation onto the modes and solves for the coefficients 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.

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 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 2110.04600 §3.3. Card: A Review of Physics-based Machine Learning in Civil Engineering.

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 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.

When to use which

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 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.

# 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 2110.04600 §3.3 2110.04600 §4.2.

Papers

• A Review of Physics-based Machine Learning in Civil Engineering: 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.