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INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solvers

Hao Wei, Aleksandra Franz, Bjoern List, Nils Thuerey

TL;DR

This work tackles the instability and error amplification of autoregressive hybrid PDE solvers that combine coarse numerical solvers with neural correctors. It introduces Indirect Neural Correction (INC), which injects the neural term as a RHS modification to the governing equations, and provides a theoretical framework showing that the resulting error growth scales as $R_k \sim \Delta t^{-1} + L$, enabling order-of-magnitude improvements in long-horizon forecasts. Through extensive experiments on KS, Burgers, and Navier–Stokes systems with diverse solvers and network backbones, INC consistently yields superior long-term accuracy, enhanced stability under aggressive coarsening, and large speedups (up to 330×) in complex 3D turbulence while preserving physics-consistent statistics. The approach is architecture- and solver-agnostic, comes with open-source code, and offers a practical pathway to fast, reliable PDE emulation with formal error-reduction guarantees.

Abstract

When simulating partial differential equations, hybrid solvers combine coarse numerical solvers with learned correctors. They promise accelerated simulations while adhering to physical constraints. However, as shown in our theoretical framework, directly applying learned corrections to solver outputs leads to significant autoregressive errors, which originate from amplified perturbations that accumulate during long-term rollouts, especially in chaotic regimes. To overcome this, we propose the Indirect Neural Corrector ($\mathrm{INC}$), which integrates learned corrections into the governing equations rather than applying direct state updates. Our key insight is that $\mathrm{INC}$ reduces the error amplification on the order of $Δt^{-1} + L$, where $Δt$ is the timestep and $L$ the Lipschitz constant. At the same time, our framework poses no architectural requirements and integrates seamlessly with arbitrary neural networks and solvers. We test $\mathrm{INC}$ in extensive benchmarks, covering numerous differentiable solvers, neural backbones, and test cases ranging from a 1D chaotic system to 3D turbulence. $\mathrm{INC}$ improves the long-term trajectory performance ($R^2$) by up to 158.7%, stabilizes blowups under aggressive coarsening, and for complex 3D turbulence cases yields speed-ups of several orders of magnitude. $\mathrm{INC}$ thus enables stable, efficient PDE emulation with formal error reduction, paving the way for faster scientific and engineering simulations with reliable physics guarantees. Our source code is available at https://github.com/tum-pbs/INC

INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solvers

TL;DR

This work tackles the instability and error amplification of autoregressive hybrid PDE solvers that combine coarse numerical solvers with neural correctors. It introduces Indirect Neural Correction (INC), which injects the neural term as a RHS modification to the governing equations, and provides a theoretical framework showing that the resulting error growth scales as , enabling order-of-magnitude improvements in long-horizon forecasts. Through extensive experiments on KS, Burgers, and Navier–Stokes systems with diverse solvers and network backbones, INC consistently yields superior long-term accuracy, enhanced stability under aggressive coarsening, and large speedups (up to 330×) in complex 3D turbulence while preserving physics-consistent statistics. The approach is architecture- and solver-agnostic, comes with open-source code, and offers a practical pathway to fast, reliable PDE emulation with formal error-reduction guarantees.

Abstract

When simulating partial differential equations, hybrid solvers combine coarse numerical solvers with learned correctors. They promise accelerated simulations while adhering to physical constraints. However, as shown in our theoretical framework, directly applying learned corrections to solver outputs leads to significant autoregressive errors, which originate from amplified perturbations that accumulate during long-term rollouts, especially in chaotic regimes. To overcome this, we propose the Indirect Neural Corrector (), which integrates learned corrections into the governing equations rather than applying direct state updates. Our key insight is that reduces the error amplification on the order of , where is the timestep and the Lipschitz constant. At the same time, our framework poses no architectural requirements and integrates seamlessly with arbitrary neural networks and solvers. We test in extensive benchmarks, covering numerous differentiable solvers, neural backbones, and test cases ranging from a 1D chaotic system to 3D turbulence. improves the long-term trajectory performance () by up to 158.7%, stabilizes blowups under aggressive coarsening, and for complex 3D turbulence cases yields speed-ups of several orders of magnitude. thus enables stable, efficient PDE emulation with formal error reduction, paving the way for faster scientific and engineering simulations with reliable physics guarantees. Our source code is available at https://github.com/tum-pbs/INC

Paper Structure

This paper contains 73 sections, 5 theorems, 97 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Neural Emulators
  4. Hybrid Neural Solvers
  5. Method: Indirect Neural Correction
  6. Theoretical analysis: Propagation of perturbations through the solver
  7. Experimental Configurations
  8. Kuramoto–Sivashinsky (KS) Equation
  9. Burgers' Equation
  10. Navier–Stokes (NS) Equation
  11. Results
  12. Accuracy in Long Autoregressive Rollouts
  13. Enhanced Accuracy for Chaotic Systems
  14. Enhanced Accuracy for Shock Waves with Reduced Resolution
  15. Improving Numerical Stability
  16. ...and 58 more sections

Key Result

Proposition 4.1

The error $\delta u^{n+1}$ at step $n+1$, arising from perturbations $\epsilon_u^n$ and $\epsilon_s^n$ at step $n$, is given by: where $J(u^n) = \frac{\partial \mathcal{N}}{\partial u}|_{u^n}$ is the Jacobian of $\mathcal{N}$ evaluated at $u^n$ and the $G(u^n) := I + \Delta t J(u^n)$, is the error amplification matrix. Proof see apd:proof_prop_local_pert.

Figures (10)

  • Figure 1: $R^2$ correlation of KS1 over 5000 steps for different methods ($\mathrm{INC}$, $\mathrm{RNN}$, $\mathrm{SITL}$, $\mathrm{SITL^*}$) applied to four models (FNO, DeepONet, UNet, ResNet). The $\mathrm{INC}$ method consistently demonstrates the most stable long-term behavior, maintaining high accuracy even beyond 5000 steps.
  • Figure 2: Top MSE of 4000-step Burgers rollouts across different neural architectures and spatial resolutions. Bottom Spatiotemporal evolution of the velocity field by different methods with 4000 steps. $\mathrm{INC}$ always performs the best
  • Figure 3: (a)KS2 simulation results with $L=21.6\pi$, $dt=0.5\text{s}$. Reference at the top, and errors of the no correction baseline and different hybrid solvers below. The solver ($\mathrm{No\ Model}$) fails after 28 steps, while hybrid solvers ($\mathrm{INC}$, $\mathrm{SITL}$, $\mathrm{SITL^*}$) continue throughout the time horizon. (b) Simulated states for the vortex street case with $dt=0.1\text{s}$. ($\mathrm{No\ Model}$) diverges within 395 steps, while hybrid solvers ($\mathrm{INC}$, $\mathrm{SITL}$, $\mathrm{SITL^*}$) remain stable, with $\mathrm{INC}$ showing the best performance.
  • Figure 4: $R^2$ correlation over time ($\frac{1}{N}\sum_{i=1}^N R^2$) for different architectures ($\mathrm{INC}$, $\mathrm{RNN}$, $\mathrm{SITL}$, $\mathrm{SITL^*}$) for both KS2 and Karman cases. All hybrid solvers improve stability compared to $\mathrm{No\ Model}$, with $\mathrm{INC}$ giving the highest correlations.
  • Figure 5: Left: BFS stream plots averaged over 1200 steps for Reference, $\mathrm{INC}$, $\mathrm{No\ Model}$, and $\mathrm{SITL}$ showing long-term emerging vortex structures in the flow. Right: efficiency-accuracy comparisons, $\mathrm{INC}$ shown in green. compared with numerical solvers of increasing resolution.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Definition 4.1: Perturbations and Errors
  • Proposition 4.1: Local Perturbations Propagation
  • Lemma 4.1: Cumulative Error
  • Definition 4.2: Error Dominance Ratio
  • Remark 4.1
  • Proposition 4.2
  • Remark 4.2
  • Theorem 4.1: Lyapunov Stability
  • Proposition 4.3: Lyapunov Stability with upper-bounds
  • Remark 4.3: Numerical Implications