想用神经网络高效求解偏微分方程？ENS从残差输入中学迭代修正，比传统混合方法快10倍，还能零样本跨方程迁移。
Error-Conditioned Neural Solvers (ENS) 将PDE残差场作为网络直接输入，让模型读取自身误差空间结构并学习迭代修正策略。在四个PDE系列上，ENS在多数设置中达到最高预测精度，在湍流Kolmogorov流上实现10倍提升。与依赖残差最小化的混合方法不同，ENS避免昂贵计算开销，并在分布偏移下泛化，包括零样本参数变化和跨方程迁移。理论证明数值最小化PDE残差在病态系统下可能不可靠，解释了现有方法的局限。
Error-Conditioned Neural Solvers
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/.