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

TL;DR

This paper introduces an adjoint-based training framework for data-driven reduced-order models that improves robustness and accuracy, especially with noisy or sparse data, by coupling operator inference with the adjoint-state method.

Contribution

It develops a novel adjoint-based training method for reduced-order models that enhances stability and accuracy over traditional approaches, particularly under noisy or limited data conditions.

Findings

The method achieves comparable accuracy to standard operator inference on clean data.

It outperforms in noisy and sparse data scenarios, providing better accuracy and stability.

Each iteration requires only one forward and one adjoint solve, making it computationally efficient.

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.