Physics-Guided Dimension Reduction for Simulation-Free Operator Learning of Stiff Differential-Algebraic Systems

TL;DR

This paper introduces a physics-informed neural network layer that exactly enforces algebraic constraints and reduces stiffness effects in simulating stiff differential-algebraic systems, improving accuracy and generalization.

Contribution

The authors develop an extended Newton implicit layer embedded in DeepONet that exactly enforces algebraic constraints and reduces stiffness effects, enabling more accurate and generalizable surrogate models for stiff DAEs.

Findings

Extended Newton layer achieves 1.42% error on a stiff inverter system.

Models compose without retraining, maintaining low error and exact constraints.

Cross-domain validation confirms the method's generalization to different stiff DAEs.

Abstract

Neural surrogates for stiff differential-algebraic equations (DAEs) face two barriers: soft-constraint methods leave algebraic residuals that stiffness amplifies into errors, and hard-constraint methods require trajectory data from stiff integrators. We introduce an extended Newton implicit layer that enforces algebraic constraints exactly and reduces fast dynamics to their quasi-steady-state values in a single differentiable solve. Embedded in a physics-informed DeepONet, the layer recovers all fast and algebraic states exactly from slow-state predictions, removes the per-window stiffness-amplification pathway, and yields a stiffness-scaled Implicit Function Theorem gradient absent from penalty methods. Cascaded implicit layers extend this to multi-component systems with provable convergence. On a grid-forming inverter (stiffness ratio of about 4712), extended Newton attains 1.42% error versus 39.3% (penalty) and 57.0% (standard Newton); augmented Lagrangian and feedback linearization diverged. Two independently trained models compose without retraining (0.72% to 1.16% error, exact constraint satisfaction). Cross-domain validation on the Robertson stiff DAE (stiffness ratio up to $10^5$) confirms generalization. Conformal prediction provides 90% coverage with automatic out-of-distribution detection.