Error-Conditioned Neural Solvers

We propose Error-Conditioned Neural Solvers (ENS), which iteratively feed the PDE residual field directly into the network to correct its own errors, achieving state-of-the-art accuracy and zero-shot extrapolation across PDE settings.

Project Overview View Results Paper

Ground Truth

ENS

FNO

PINO

DiffusionPDE

PCFM

Solution

PDE residual

L1 error

Project pipeline overview — **ENS pipeline.** ENS uses the network \( \mathcal{P}_\theta \) for the initial prediction \( \hat{u}_{(0)} \), followed by \( k \) recurrent refinement steps via the corrector network \( \mathcal{C}_\phi(f, \hat{u}_{(t)}, r_{(t)}) \) that takes the current residual \( r_{(t)} \) of \( \hat{u}_{(t)} \) as its input.

View Abstract

Neural operators learn fast approximate mappings from PDE parameters to solutions. Recent hybrid methods combine neural operators with external numerical corrections, e.g., gradient descent or Gauss-Newton steps on the PDE residual, to improve physical accuracy, but inherit the instability, cost, and pitfalls of classical solvers. In this work, we propose Error-Conditioned Neural Solvers (ENS), which internalize the correction within the network itself. Our neural solvers recurrently correct its predictions by computing the PDE residual field of the current prediction and passing it as an explicit input to the network, until the residual converges to a consistent floor. ENS achieves state-of-the-art prediction accuracy across PDE families while matching the PDE residuals of hybrid optimization methods without their instability or compute cost. We theoretically and empirically show that minimizing PDE residual is an unreliable proxy for reconstruction accuracy for ill-conditioned systems, explaining why hybrid methods fail under distribution shift despite low residuals — a regime where ENS maintains accuracy across diverse extrapolation settings, including zero-shot parameter shifts, super-resolution, and cross-equation transfer.