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DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction

Rudy Morel, Jiequn Han, Edouard Oyallon

TL;DR

DISCO tackles next-state prediction for PDE-governed systems with unknown dynamics by learning an evolution operator from context via a transformer-based hypernetwork that outputs a compact operator for a neural-ODE-like solver. This decouples dynamics inference from state evolution and embeds a finite-difference–style inductive bias, enabling data-efficient pretraining across diverse PDEs. Across PDEBench and The Well, DISCO achieves state-of-the-art or competitive next-step predictions with far fewer training epochs and shows robust generalization to unseen physics and fine-tuning scenarios. The approach suggests a scalable path for physics-informed meta-learning and points to extensions such as graph-based operators for nonuniform meshes and time-aware hypernetworks for time-dependent dynamics.

Abstract

We address the problem of predicting the next state of a dynamical system governed by unknown temporal partial differential equations (PDEs) using only a short trajectory. While standard transformers provide a natural black-box solution to this task, the presence of a well-structured evolution operator in the data suggests a more tailored and efficient approach. Specifically, when the PDE is fully known, classical numerical solvers can evolve the state accurately with only a few parameters. Building on this observation, we introduce DISCO, a model that uses a large hypernetwork to process a short trajectory and generate the parameters of a much smaller operator network, which then predicts the next state through time integration. Our framework decouples dynamics estimation (i.e., DISCovering an evolution operator from a short trajectory) from state prediction (i.e., evolving this operator). Experiments show that pretraining our model on diverse physics datasets achieves state-of-the-art performance while requiring significantly fewer epochs. Moreover, it generalizes well and remains competitive when fine-tuned on downstream tasks.

DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction

TL;DR

DISCO tackles next-state prediction for PDE-governed systems with unknown dynamics by learning an evolution operator from context via a transformer-based hypernetwork that outputs a compact operator for a neural-ODE-like solver. This decouples dynamics inference from state evolution and embeds a finite-difference–style inductive bias, enabling data-efficient pretraining across diverse PDEs. Across PDEBench and The Well, DISCO achieves state-of-the-art or competitive next-step predictions with far fewer training epochs and shows robust generalization to unseen physics and fine-tuning scenarios. The approach suggests a scalable path for physics-informed meta-learning and points to extensions such as graph-based operators for nonuniform meshes and time-aware hypernetworks for time-dependent dynamics.

Abstract

We address the problem of predicting the next state of a dynamical system governed by unknown temporal partial differential equations (PDEs) using only a short trajectory. While standard transformers provide a natural black-box solution to this task, the presence of a well-structured evolution operator in the data suggests a more tailored and efficient approach. Specifically, when the PDE is fully known, classical numerical solvers can evolve the state accurately with only a few parameters. Building on this observation, we introduce DISCO, a model that uses a large hypernetwork to process a short trajectory and generate the parameters of a much smaller operator network, which then predicts the next state through time integration. Our framework decouples dynamics estimation (i.e., DISCovering an evolution operator from a short trajectory) from state prediction (i.e., evolving this operator). Experiments show that pretraining our model on diverse physics datasets achieves state-of-the-art performance while requiring significantly fewer epochs. Moreover, it generalizes well and remains competitive when fine-tuned on downstream tasks.
Paper Structure (47 sections, 15 equations, 26 figures, 7 tables)

This paper contains 47 sections, 15 equations, 26 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related works
  4. Neural solvers and operators.
  5. Methodology
  6. Preliminaries
  7. Finite difference and neural PDEs.
  8. Method: DISCovering an Operator from context
  9. Our framework.
  10. Numerical integration.
  11. Operator network architecture $f_\theta$.
  12. Hypernetwork architecture $\psi$.
  13. End-to-end training.
  14. Numerical experiments
  15. Multiple physics datasets.
  16. ...and 32 more sections

Figures (26)

  • Figure 1: (a) A task-specific learned operator must be re-trained on unseen Physics. (b) A transformer (e.g., MPP) must learn both an encoder and a decoder, adding reconstruction error. (c) Our DISCO model infers operator parameters without relying on an auto-encoder.
  • Figure 2: Burgers
  • Figure 3: Shallow water
  • Figure 4: Diff.-reaction
  • Figure 5: Incomp. NS
  • ...and 21 more figures