[NeurIPS 2025] OpsSplit: Moving from Learning Solutions to Learning Physical Operators

TL;DR

Current neural PDE solvers struggle with "out-of-distribution" (OOD) scenarios because they try to learn a monolithic mapping from state A to state B. OpsSplit changes the paradigm: it decomposes the PDE into its constituent physical parts (convection, diffusion, etc.), using a Mixture of Experts (MoE) approach to learn non-linear operators while hard-coding linear ones. The result? A model that doesn't just predict the next frame—it understands the underlying physics well enough to generalize to unseen viscosities and geometries.

The Blind Spot of Modern Neural Operators

Most Neural Operators (NOs) are "Solution Operators." They treat a PDE like a video prediction task—mapping $u(t)$ to $u(t+dt)$. While efficient, this approach has a fatal flaw: spectral bias and entanglement. A single neural network must learn high-frequency turbulence and low-frequency diffusion simultaneously. When you change a physical parameter (like viscosity $u$), these monolithic models break because the physics they "memorized" no longer applies.

The OpsSplit Insight: Physics is Modular

The authors propose OpsSplit, which mimics classical numerical techniques by splitting the spatial operator $D_X(u)$ into linear and non-linear components.

Why this works:

1. Disentanglement: Instead of learning a black box, the model assigns one "Expert" neural network to learn the non-linear convection term $(\mathbf{v}\cdot abla)\mathbf{v}$ and uses a simple, fixed convolution (finite difference) for linear diffusion $uabl{a}^2\mathbf{v}$.
2. Neural ODE Backbone: These split operators form the right-hand side (RHS) of an ODE. By integrating this RHS, the model can predict state changes continuously in time, rather than in fixed discrete steps.

Figure 1: Traditional Autoregressive vs. Neural ODE vs. the proposed OpsSplit approach.

Performance: Crushing the OOD Barrier

The true test for any PDE solver is Temporal Extrapolation (predicting far beyond the training horizon) and OOD Parameter Shifts (changing physics constants).

Key Experimental Findings:

• Incompressible Navier-Stokes: In OOD extrapolation tests, OpsSplit using an FNO backbone achieved an NRMSE of 0.3282, compared to a staggering 0.9535 for traditional solution-mapping models.
• Data Efficiency: Because the model "knows" part of the physics (the linear kernels), it requires significantly less data to reach high accuracy.
• Interpretability: You can actually visualize the learned operators! The authors compared the "Latent Convection" learned by FNO with a ground-truth numerical convection—the patterns are strikingly similar, proving the model is learning real physical gradients.

Figure 2: Rollout error growth on OOD scenarios. OpsSplit (Physical) maintains stability while others diverge.

Cross-PDE Transfer Learning: The "Holy Grail"

Perhaps the most exciting result is bidirectional transfer learning. The authors found that a convection operator trained on Compressible Navier-Stokes could be "plugged into" an Incompressible system. This fine-tuning led to faster convergence and lower test error than training from scratch. This modularity suggests a future where we have "libraries" of pre-trained physical operators that can be assembled to solve entirely new multi-physics problems.

Limitations & Future Work

While powerful, OpsSplit isn't a free lunch:

• Domain Expertise Required: You must know the PDE form to perform the split.
• Computational Cost: Neural ODEs require more forward passes during integration than simple autoregressive rollouts.
• Topology: Stretching this to irregular, non-rectangular grids (using Graph Neural Operators) is possible but adds complexity to the finite-difference kernels.

Conclusion

OpsSplit represents a significant step towards AI-Physicist models. By respecting the modular nature of partial differential equations, we can build surrogate models that are not just accurate, but robust, interpretable, and transferable across the vast landscape of scientific computing.

Senior Editor’s Note: This work elegantly bridges the gap between the "Optimize-then-Discretize" (PINNs) and "Discretize-then-Optimize" (NOs) paradigms. It proves that a little bit of structural induct bias—knowing that physics is a sum of operators—goes a long way in solving the OOD bottleneck.