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Space and Time Continuous Physics Simulation From Partial Observations

Janny Steeven, Nadri Madiha, Digne Julie, Wolf Christian

TL;DR

This work addresses the challenge of simulating physical systems governed by unknown PDEs from sparse observations while enabling queries at arbitrary spatial and temporal locations. It introduces a double dynamical system framework: System 1 performs auto-regressive forecasting of latent anchor states in a mesh-based graph, and System 2 uses a spatio-temporal attention-based observer to reconstruct the dense field S($\boldsymbol{s}_0,\boldsymbol{x},t$) at any $(\boldsymbol{x},t)$. The authors provide theoretical bounds showing latent-space forecasting can yield tighter error control than classic autoregression on the observed data and prove the existence of a finite-horizon observer with a provable error bound. Empirically, the method outperforms strong baselines (e.g., MeshGraphNet, MaGNet, DINo) on three challenging CFD datasets, demonstrating robust space-time interpolation, extrapolation to unseen grids, and generalization to new initial conditions. This approach enables accurate, flexible, and scalable continuous simulations and has potential applications beyond fluid dynamics, while relying on observability and sampling assumptions that can guide future physics-informed enhancements.

Abstract

Modern techniques for physical simulations rely on numerical schemes and mesh-refinement methods to address trade-offs between precision and complexity, but these handcrafted solutions are tedious and require high computational power. Data-driven methods based on large-scale machine learning promise high adaptivity by integrating long-range dependencies more directly and efficiently. In this work, we focus on fluid dynamics and address the shortcomings of a large part of the literature, which are based on fixed support for computations and predictions in the form of regular or irregular grids. We propose a novel setup to perform predictions in a continuous spatial and temporal domain while being trained on sparse observations. We formulate the task as a double observation problem and propose a solution with two interlinked dynamical systems defined on, respectively, the sparse positions and the continuous domain, which allows to forecast and interpolate a solution from the initial condition. Our practical implementation involves recurrent GNNs and a spatio-temporal attention observer capable of interpolating the solution at arbitrary locations. Our model not only generalizes to new initial conditions (as standard auto-regressive models do) but also performs evaluation at arbitrary space and time locations. We evaluate on three standard datasets in fluid dynamics and compare to strong baselines, which are outperformed both in classical settings and in the extended new task requiring continuous predictions.

Space and Time Continuous Physics Simulation From Partial Observations

TL;DR

This work addresses the challenge of simulating physical systems governed by unknown PDEs from sparse observations while enabling queries at arbitrary spatial and temporal locations. It introduces a double dynamical system framework: System 1 performs auto-regressive forecasting of latent anchor states in a mesh-based graph, and System 2 uses a spatio-temporal attention-based observer to reconstruct the dense field S() at any . The authors provide theoretical bounds showing latent-space forecasting can yield tighter error control than classic autoregression on the observed data and prove the existence of a finite-horizon observer with a provable error bound. Empirically, the method outperforms strong baselines (e.g., MeshGraphNet, MaGNet, DINo) on three challenging CFD datasets, demonstrating robust space-time interpolation, extrapolation to unseen grids, and generalization to new initial conditions. This approach enables accurate, flexible, and scalable continuous simulations and has potential applications beyond fluid dynamics, while relying on observability and sampling assumptions that can guide future physics-informed enhancements.

Abstract

Modern techniques for physical simulations rely on numerical schemes and mesh-refinement methods to address trade-offs between precision and complexity, but these handcrafted solutions are tedious and require high computational power. Data-driven methods based on large-scale machine learning promise high adaptivity by integrating long-range dependencies more directly and efficiently. In this work, we focus on fluid dynamics and address the shortcomings of a large part of the literature, which are based on fixed support for computations and predictions in the form of regular or irregular grids. We propose a novel setup to perform predictions in a continuous spatial and temporal domain while being trained on sparse observations. We formulate the task as a double observation problem and propose a solution with two interlinked dynamical systems defined on, respectively, the sparse positions and the continuous domain, which allows to forecast and interpolate a solution from the initial condition. Our practical implementation involves recurrent GNNs and a spatio-temporal attention observer capable of interpolating the solution at arbitrary locations. Our model not only generalizes to new initial conditions (as standard auto-regressive models do) but also performs evaluation at arbitrary space and time locations. We evaluate on three standard datasets in fluid dynamics and compare to strong baselines, which are outperformed both in classical settings and in the extended new task requiring continuous predictions.
Paper Structure (19 sections, 2 theorems, 35 equations, 10 figures, 5 tables)

This paper contains 19 sections, 2 theorems, 35 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Continuous Solutions from Sparse Observations
  4. The double observation problem
  5. Theoretical analysis
  6. Implementation
  7. Training
  8. Experimental Results
  9. Conclusion
  10. Acknowledgements
  11. Website and interactive online visualization
  12. Proof of proposition \ref{['prop:neurips_bounds']}
  13. Comparison of upper bounds in Proposition \ref{['prop:neurips_bounds']}
  14. Proof of proposition \ref{['prop:neurips_observer']}
  15. Model description
  16. ...and 4 more sections

Key Result

Proposition 1

Consider a dynamical system of the form of System 1 and assume the existence of a state observer $\textcolor{coolblue}{e}$ along with approximations $\textcolor{coolgreen}{\hat{f_1}}, \textcolor{coolgreen}{\hat{h_1}}, \textcolor{coolgreen}{\hat{e}}$ with Lipschitz constants $L_f, L_h$ and $L_e$ resp for the Euclidean norm $|\cdot|$, then for all integer $n>0$, with $\hat{{\bm{s}}}_d[n]$ and $\hat{

Figures (10)

  • Figure 1: Model overview -- We achieve space and time continuous simulations of physics systems by formulating the task as a double observation problem. System 1 is a discrete dynamical model used to compute a sequence of latent anchor states ${\bm{z}}_d$ auto-regressively, and System 2 is used to design a state estimator $\psi_q$ retrieving the dense physical state at arbitrary locations $({\bm{x}}, t)$.
  • Figure 2: Results on Eagle -- Per point error of the flow prediction on an Eagle example in the Low spatial down-sampling scenario. Our model exhibits lower errors as also shown in Tables \ref{['tab:space_continuity']} and \ref{['tab:time_continuity']}.
  • Figure 3: Model overview -- The model leverages a dynamical system (System 1) to perform auto-regressive predictions of the dynamics in a mesh-structured latent space from sparse initial conditions. It is combined with a data-driven state estimator derived from another continuous-time dynamical system (System 2), implemented with multi-head cross-attention. The attention mechanism queries the intermediate anchor states from the auto-regressive predictor and uses Fourier positional encoding to encode the query points $(\mathbf{x},\tau)$. An additional GRU refines the dynamics after interpolation.
  • Figure 4: MaGNet -- suffers from drastic shifts in distribution between training and evaluation. The model is trained on points from ${\mathcal{X}}$, which corresponds to a small portion of the domain. We used our subsampling trick to artificially generate queries. During evaluation, we require the prediction at every available point in the complete simulation, hence, MaGNet must interpolate the initial condition to a large number of query points, filling the input of the auto-regressive model with noisy estimates of the IC.
  • Figure 5: Qualitative results on Shallow-Water -- Simulation obtained with our model and the baseline in the challenging 5% setup on the Shallow Water dataset (without temporal sub-sampling). Each model is initialized with a small set of sparse observations and needs to extrapolate the solution at many unseen positions. Our model outperforms the baselines, which struggle to compute the solution outside the training domain.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2