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An adjoint method for training data-driven reduced-order models

Donglin Liu, Francisco García Atienza, Mengwu Guo

TL;DR

This work addresses the vulnerability of derivative-based operator inference (OpInf) to noisy and sparsely sampled data by introducing a continuous-time, adjoint-based training framework. It reframes learning as trajectory fitting, using a backward adjoint to compute exact gradients with respect to reduced operators, and employs a short-horizon, multiple-shooting strategy to stabilize long-time training. Across three canonical PDEs, the adjoint-trained quadratic ROMs achieve comparable performance to OpInf on clean data but offer markedly better robustness to sampling sparsity and additive noise, often preserving stability where high-order finite-difference OpInf degrades. The approach provides a practical, scalable pathway to robust data-driven ROMs for large-scale simulations with imperfect training data, with potential extensions to structure preservation and parametric variability.

Abstract

Reduced-order modeling lies at the interface of numerical analysis and data-driven scientific computing, providing principled ways to compress high-fidelity simulations in science and engineering. We propose a training framework that couples a continuous-time form of operator inference with the adjoint-state method to obtain robust data-driven reduced-order models. This method minimizes a trajectory-based loss between reduced-order solutions and projected snapshot data, which removes the need to estimate time derivatives from noisy measurements and provides intrinsic temporal regularization through time integration. We derive the corresponding continuous adjoint equations to compute gradients efficiently and implement a gradient based optimizer to update the reduced model parameters. Each iteration only requires one forward reduced order solve and one adjoint solve, followed by inexpensive gradient assembly, making the method attractive for large-scale simulations. We validate the proposed method on three partial differential equations: viscous Burgers' equation, the two-dimensional Fisher-KPP equation, and an advection-diffusion equation. We perform systematic comparisons against standard operator inference under two perturbation regimes, namely reduced temporal snapshot density and additive Gaussian noise. For clean data, both approaches deliver similar accuracy, but in situations with sparse sampling and noise, the proposed adjoint-based training provides better accuracy and enhanced roll-out stability.

An adjoint method for training data-driven reduced-order models

TL;DR

This work addresses the vulnerability of derivative-based operator inference (OpInf) to noisy and sparsely sampled data by introducing a continuous-time, adjoint-based training framework. It reframes learning as trajectory fitting, using a backward adjoint to compute exact gradients with respect to reduced operators, and employs a short-horizon, multiple-shooting strategy to stabilize long-time training. Across three canonical PDEs, the adjoint-trained quadratic ROMs achieve comparable performance to OpInf on clean data but offer markedly better robustness to sampling sparsity and additive noise, often preserving stability where high-order finite-difference OpInf degrades. The approach provides a practical, scalable pathway to robust data-driven ROMs for large-scale simulations with imperfect training data, with potential extensions to structure preservation and parametric variability.

Abstract

Reduced-order modeling lies at the interface of numerical analysis and data-driven scientific computing, providing principled ways to compress high-fidelity simulations in science and engineering. We propose a training framework that couples a continuous-time form of operator inference with the adjoint-state method to obtain robust data-driven reduced-order models. This method minimizes a trajectory-based loss between reduced-order solutions and projected snapshot data, which removes the need to estimate time derivatives from noisy measurements and provides intrinsic temporal regularization through time integration. We derive the corresponding continuous adjoint equations to compute gradients efficiently and implement a gradient based optimizer to update the reduced model parameters. Each iteration only requires one forward reduced order solve and one adjoint solve, followed by inexpensive gradient assembly, making the method attractive for large-scale simulations. We validate the proposed method on three partial differential equations: viscous Burgers' equation, the two-dimensional Fisher-KPP equation, and an advection-diffusion equation. We perform systematic comparisons against standard operator inference under two perturbation regimes, namely reduced temporal snapshot density and additive Gaussian noise. For clean data, both approaches deliver similar accuracy, but in situations with sparse sampling and noise, the proposed adjoint-based training provides better accuracy and enhanced roll-out stability.
Paper Structure (39 sections, 1 theorem, 60 equations, 12 figures, 2 algorithms)

This paper contains 39 sections, 1 theorem, 60 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Reduced-order models via operator inference
  3. The adjoint method for operator inference
  4. Continuous-time loss functional
  5. The adjoint method
  6. Derivative notation
  7. Quadratic ROM: explicit derivatives
  8. Computational implementation
  9. Numerical experiments
  10. Perturbation protocols
  11. Baseline and initialization (OpInf)
  12. Training regime (adjoint)
  13. Optimization and numerics
  14. Viscous burgers' equation
  15. Fisher-KPP equation
  16. ...and 24 more sections

Key Result

Proposition 1

Assume $\mathbf f(\mathbf q;\bm\theta)$ is continuously differentiable in both $\mathbf q$ and $\bm\theta$ and that the initial condition $\mathbf q_0$ is independent of $\bm\theta$. Let $\tilde{\mathbf q}(\cdot;\bm\theta)$ solve the state equation in eq:opt_problem. The gradient of the reduced loss where the adjoint state variable $\bm\lambda:[0,T]\to\mathbb R^r$ satisfies the backward-in-time OD

Figures (12)

  • Figure 1: Numerical solution of the viscous Burgers’ equation. Space–time contour of $u(x,t_i)$.
  • Figure 2: Model performance on Burgers’ equation under varying noise and sampling. Columns vary the noise level (NL = 0–200% of the state standard deviation); rows vary the number of snapshots across train, validation, and test (20, 100, 1000, 10000). Each panel shows test RSE ($\log_{10}$) versus ROM dimension $r$. Methods: Adjoint (ours), OpInf-ord2, and OpInf-ord6. Lower is better.
  • Figure 3: Reduced-coordinate rollouts for viscous Burgers’ equation on the test window. ROM dimension $r=5$; $1000$ total snapshots across train, validation, and test. Rows vary the noise level (NL = 0%, 80%, 160%); columns compare OpInf-ord2, OpInf-ord6, and the adjoint method (ours). Gray vertical dashed lines mark train/validation/test splits. Colored markers are the noisy observations $\mathbf q_{\text{true}}(t)$ used for training/validation; solid curves are the clean reference trajectory; cyan dashed curves are model predictions initialized at the test initial condition. Under heavy noise, the adjoint rollout follows the clean trajectory more closely, especially beyond the training window.
  • Figure 4: Spatiotemporal evolution $u(x,t)$ for viscous Burgers’ equation on the test window. ROM dimension $r=5$; $1000$ total snapshots across train, validation, and test. Rows vary NL = 0%, 80%, 160%; columns show the FOM and the corresponding reconstruction using POD, OpInf-ord2, OpInf-ord6, and the adjoint ROM. Color maps share a panel-wise scale. The adjoint model best preserves the location and amplitude of the evolving profile under noise, while OpInf-ord6 exhibits pronounced noise imprinting and OpInf-ord2 shows bias and smoothing.
  • Figure 5: Numerical solution of Fisher-KPP equation. Space–time contour of $u(x,t_i)$.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Proposition 1: The adjoint method
  • proof
  • proof