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Breaking the Discretization Barrier of Continuous Physics Simulation Learning

Fan Xu, Hao Wu, Nan Wang, Lilan Peng, Kun Wang, Wei Gong, Xibin Zhao

TL;DR

CoPS tackles the challenge of learning continuous spatiotemporal physics from partial, unstructured observations by combining a multiplicative filter network encoding with customized geometric grids, a multi-scale graph ODE for continuous-time latent dynamics, and a neural auto-correction module that enforces discrete refinements. The approach enables grid-independent querying and robust long-horizon predictions, demonstrated across synthetic and real-world fluid and atmospheric datasets with superior accuracy over strong baselines. Key contributions include a novel encoding mechanism on irregular grids, a hierarchical graph-ODE dynamics module, and a Markov-based correction scheme with theoretical stability guarantees. The results suggest substantial potential for accurate, scalable, and continuous physical forecasting in geophysics and related domains, albeit with acknowledged computational costs and a simplifying Markov assumption in corrections.

Abstract

The modeling of complicated time-evolving physical dynamics from partial observations is a long-standing challenge. Particularly, observations can be sparsely distributed in a seemingly random or unstructured manner, making it difficult to capture highly nonlinear features in a variety of scientific and engineering problems. However, existing data-driven approaches are often constrained by fixed spatial and temporal discretization. While some researchers attempt to achieve spatio-temporal continuity by designing novel strategies, they either overly rely on traditional numerical methods or fail to truly overcome the limitations imposed by discretization. To address these, we propose CoPS, a purely data-driven methods, to effectively model continuous physics simulation from partial observations. Specifically, we employ multiplicative filter network to fuse and encode spatial information with the corresponding observations. Then we customize geometric grids and use message-passing mechanism to map features from original spatial domain to the customized grids. Subsequently, CoPS models continuous-time dynamics by designing multi-scale graph ODEs, while introducing a Markov-based neural auto-correction module to assist and constrain the continuous extrapolations. Comprehensive experiments demonstrate that CoPS advances the state-of-the-art methods in space-time continuous modeling across various scenarios.

Breaking the Discretization Barrier of Continuous Physics Simulation Learning

TL;DR

CoPS tackles the challenge of learning continuous spatiotemporal physics from partial, unstructured observations by combining a multiplicative filter network encoding with customized geometric grids, a multi-scale graph ODE for continuous-time latent dynamics, and a neural auto-correction module that enforces discrete refinements. The approach enables grid-independent querying and robust long-horizon predictions, demonstrated across synthetic and real-world fluid and atmospheric datasets with superior accuracy over strong baselines. Key contributions include a novel encoding mechanism on irregular grids, a hierarchical graph-ODE dynamics module, and a Markov-based correction scheme with theoretical stability guarantees. The results suggest substantial potential for accurate, scalable, and continuous physical forecasting in geophysics and related domains, albeit with acknowledged computational costs and a simplifying Markov assumption in corrections.

Abstract

The modeling of complicated time-evolving physical dynamics from partial observations is a long-standing challenge. Particularly, observations can be sparsely distributed in a seemingly random or unstructured manner, making it difficult to capture highly nonlinear features in a variety of scientific and engineering problems. However, existing data-driven approaches are often constrained by fixed spatial and temporal discretization. While some researchers attempt to achieve spatio-temporal continuity by designing novel strategies, they either overly rely on traditional numerical methods or fail to truly overcome the limitations imposed by discretization. To address these, we propose CoPS, a purely data-driven methods, to effectively model continuous physics simulation from partial observations. Specifically, we employ multiplicative filter network to fuse and encode spatial information with the corresponding observations. Then we customize geometric grids and use message-passing mechanism to map features from original spatial domain to the customized grids. Subsequently, CoPS models continuous-time dynamics by designing multi-scale graph ODEs, while introducing a Markov-based neural auto-correction module to assist and constrain the continuous extrapolations. Comprehensive experiments demonstrate that CoPS advances the state-of-the-art methods in space-time continuous modeling across various scenarios.

Paper Structure

This paper contains 31 sections, 1 theorem, 27 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Deep Learning for Physical Simulations
  4. Advanced Space-Continuous Modeling
  5. Methodology
  6. Initial Encoding via Multiplicative Filter Network
  7. Latent Dynamics Modeling
  8. Decoder
  9. Theoretical Analysis
  10. Experiment
  11. Experimental Settings
  12. Main Results
  13. Model Analysis
  14. Conclusion
  15. Detailed Description of Datasets
  16. ...and 16 more sections

Key Result

Theorem 3.1

Consider a spatiotemporal dynamical system whose latent representation $y(t)$ is intended to follow an ideal trajectory $y^*(t)$. The learned dynamics are modeled by a Neural ODE: with initial condition $y(t_0)$. The function $\mathcal{F}$, representing the multi-scale graph message passing, is assumed to be $L_{\mathcal{F}}$-Lipschitz continuous with respect to $y(t)$. Periodic corrections are a

Figures (7)

  • Figure 1: Example of continuous time and space dynamic system evolution. (Non-discretized)
  • Figure 2: The overview of CoPS. Stage 1: Employ multiplicative filter network to encode the initial representation, and map it to the customized grids with message passing scheme. Stage 2: Model the latent dynamics with multi-scale Graph ODE module and auto-correction module in a continuous-time way. Stage 3: Extrapolate results for arbitrary future time step and coordinates.
  • Figure 3: Figure(a) and (b) shows the perdiction performance based on observations with diverse ratio of subsampling (25%, 50%, 75%) on the Navier-Stokes and Kuroshio dataset. Figure(c) demonstrates long-term extrapolation beyond the training horizon on the Prometheus dataset, and Figure(e) shows the evolution of prediction errors over time steps, revealing the increasing error with longer prediction steps. Figure(d) illustrates continuous-time prediction for intermediate points between discrete time steps on the WeatherBench dataset.
  • Figure 4: Left shows the inference performance for super-resolution on Navier-Stokes dataset. Right demonstrates the resolution generalization capability compared with MAgNet, DINo, and ContiPDE.
  • Figure 5: Qualitative comparison of our method and ContiPDE on the Navier-Stokes dataset under sparse initial observations (25% and 75%). In each panel, the first row displays the ground truth evolution, spanning both in-horizon (training) and out-horizon (extrapolation) timesteps. The second and third rows depict the predictions generated by our proposed CoPS and ContiPDE.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 3.1: Error Bounding via Hybrid Continuous-Discrete Latent Corrections