Error-Conditioned Neural Solvers: Hybrid Physics-Corrected PDE Surrogates

Forum topic · 小凯 · 2026-06-27

Summary

This arXiv paper (2606.27354) by Haina Jiang, Liam Wang, and Peng-Chen Chen, published 2026-06-27, addresses a key weakness of neural surrogate models for PDE solving. Standard surrogates map PDE parameters to solutions quickly but treat solving as a purely statistical task, so after training they struggle to correct their own constraint violations and fail to extrapolate beyond the training distribution. Recent hybrid approaches improve physical correctness by minimizing the PDE residual with gradient descent or Gauss–Newton steps, but they inherit the computational cost and instability of classical optimizers. The paper proposes error-conditioned neural solvers and supports the approach with theoretical analysis, aiming to combine fast surrogate inference with reliable physics correction at lower cost. Full abstract and details are available on arXiv.

Paper Overview

Field: Computer Vision (CV) Authors: Haina Jiang, Liam Wang, Peng-Chen Chen Published: 2026-06-27 arXiv: 2606.27354

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. The authors show, theoretically and empirically, how conditioning the solver on its own error can address these limitations.

Key Points

Neural PDE surrogates are fast but statistically driven; they cannot reliably self-correct constraint violations or extrapolate out of distribution.
Hybrid residual-correction methods (gradient descent, Gauss–Newton) improve physical fidelity but remain expensive and unstable.
The proposed error-conditioned approach aims to combine surrogate speed with physics-based correctness at reduced cost.

> Note: The original forum post contains a truncated abstract ("We show, theore..."). See the full paper on arXiv for complete theoretical results and experiments.

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*Auto-collected on 2026-06-27*