# Space and Time Continuous Physics Simulation From Partial Observations

arXiv: https://arxiv.org/abs/2401.09198 · HTML: https://mlsanity.com/paper/2401.09198/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 49 references that are in the map.
- First posted on arXiv 17 Jan 2024.

## Builds on (most cited first, 30 of 31)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids](https://mlsanity.com/paper/1810.01566/) (2018)
- [Disentangling Physical Dynamics from Unknown Factors for Unsupervised Video Prediction](https://mlsanity.com/paper/2003.01460/) (2020)
- [Translating Images into Maps](https://mlsanity.com/paper/2110.00966/) (2021)
- [Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge](https://mlsanity.com/paper/1711.07970/) (2017)
- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [Continuous PDE Dynamics Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2209.14855/) (2022)
- [Eagle: Large-Scale Learning of Turbulent Fluid Dynamics with Mesh Transformers](https://mlsanity.com/paper/2302.10803/) (2023)
- [Learning to Accelerate Partial Differential Equations via Latent Global Evolution](https://mlsanity.com/paper/2206.07681/) (2022)
- [MAgNet: Mesh Agnostic Neural PDE Solver](https://mlsanity.com/paper/2210.05495/) (2022)
- [Competitive Physics Informed Networks](https://mlsanity.com/paper/2204.11144/) (2022)
- [A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks](https://mlsanity.com/paper/2304.14994/) (2023)
- [Filtered-CoPhy: Unsupervised Learning of Counterfactual Physics in Pixel Space](https://mlsanity.com/paper/2202.00368/) (2022)
- [Simple initialization and parametrization of sinusoidal networks via their kernel bandwidth](https://mlsanity.com/paper/2211.14503/) (2022)

## Built on by (most cited first, 6 of 6)

- [Spatio-Temporal Fluid Dynamics Modeling via Physical-Awareness and Parameter Diffusion Guidance](https://mlsanity.com/paper/2403.13850/) (2024)
- [Reasoning in visual navigation of end-to-end trained agents: a dynamical systems approach](https://mlsanity.com/paper/2503.08306/) (2025)
- [Breaking the Discretization Barrier of Continuous Physics Simulation Learning](https://mlsanity.com/paper/2509.17955/) (2025)
- [A Two-Phase Deep Learning Framework for Adaptive Time-Stepping in High-Speed Flow Modeling](https://mlsanity.com/paper/2506.07969/) (2025)
- [Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reduction](https://mlsanity.com/paper/2407.03925/) (2024)
- [LASER: Learning Active Sensing for Continuum Field Reconstruction](https://mlsanity.com/paper/2604.19355/) (2026)

## Abstract (arXiv)

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.