Error-Conditioned Neural Solvers

2026-06-25

AI Summary

Traditional neural surrogate models struggle to correct violations of partial differential equation (PDE) constraints and exhibit limited out-of-distribution generalization. Existing hybrid approaches based on residual minimization are computationally expensive and unstable, particularly in ill-posed systems where small residuals do not guarantee accurate solutions. This work proposes the Error-conditioned Neural Solver (ENS), which abandons the paradigm of using residuals as optimization objectives and instead explicitly encodes them as network inputs for the first time. This enables the model to perceive the structure of the error space and learn iterative correction strategies. ENS employs an end-to-end trainable architecture that requires no external optimizer and achieves state-of-the-art accuracy across four PDE families—reducing errors by an order of magnitude on turbulent Kolmogorov flow—while supporting zero-shot parameter variations and cross-equation transfer, significantly enhancing robustness and accuracy under ill-posed conditions and distribution shifts.

Abstract

Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching $10\times$ on turbulent Kolmogorov flow, while avoiding the expensive compute cost of hybrid methods. ENS's learned correction policy generalizes under distribution shift, including zero-shot parameter changes and cross-equation transfer, where its relative advantage is largest in the ill-conditioned regimes where residual minimization is least reliable. Project website: https://neuralsolver.github.io/.

Problem

• neural solvers
• PDE residual
• ill-conditioned systems
• physical constraints
• distribution shift

Innovation

• Error-Conditioned Neural Solvers
• PDE residual as input
• iterative correction policy
• generalization under distribution shift
• ill-conditioned PDEs