Error-Conditioned Neural Solvers
2026-06-25 • Machine Learning
Machine LearningArtificial IntelligenceComputer Vision and Pattern Recognition
AI summaryⓘ
The authors studied neural networks that quickly approximate solutions to complex math equations (PDEs). They found that just minimizing error measurements (PDE residuals) doesn’t always mean the solution is accurate, especially for tricky problems. So, they created a new method called ENS that feeds error information back into the network so it can learn how to fix its own mistakes step-by-step. This approach works better than older methods, especially for difficult cases, and doesn’t need as much computing power. Their method also adapts well to changes in the problem or different equations.
Partial Differential Equations (PDEs)Neural surrogate modelsPDE residualGradient descentGauss–Newton methodIll-conditioned systemsError feedbackIterative correctionDistribution shiftKolmogorov flow
Authors
Haina Jiang, Liam Wang, Peng-Chen Chen, Min Seop Kwak, Seungryong Kim, Brian Bell, Jeong Joon Park
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/.