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

2026-01-12

AI Summary

This work proposes a novel framework that integrates continuous-time operator inference with the adjoint-state method to address the poor accuracy and unstable extrapolation of traditional data-driven reduced-order models under sparse sampling and noisy data. By minimizing trajectory loss during training, the approach avoids direct differentiation of noisy measurements and leverages temporal integration for intrinsic regularization. For the first time, the adjoint method is incorporated into continuous-time operator inference, enabling efficient gradient computation and stable optimization. Combining continuous adjoint equations, projected snapshot matching, and gradient-based optimization, the method demonstrates significantly improved accuracy and rolling prediction stability over standard operator inference when tested on the Burgers, Fisher–KPP, and convection–diffusion equations under sparse or noisy data conditions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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.

Problem

• reduced-order modeling
• data-driven
• noise robustness
• trajectory-based loss
• operator inference

Innovation

• adjoint method
• operator inference
• reduced-order modeling
• trajectory-based loss
• data-driven scientific computing