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Learning to Accelerate Partial Differential Equations via Latent Global Evolution

arXiv 2206.07681 ↗First posted 15 Jun 2022cs.LG physics.comp-phcited by 39

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Abstract (arXiv)

Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems. However, both classical solvers and recent deep learning-based surrogate models are typically extremely computationally intensive, because of their local evolution: they need to update the state of each discretized cell at each time step during inference. Here we develop Latent Evolution of PDEs (LE-PDE), a simple, fast and scalable method to accelerate the simulation and inverse optimization of PDEs. LE-PDE learns a compact, global representation of the system and efficiently evolves it fully in the latent space with learned latent evolution models. LE-PDE achieves speed-up by having a much smaller latent dimension to update during long rollout as compared to updating in the input space. We introduce new learning objectives to effectively learn such latent dynamics to ensure long-term stability. We further introduce techniques for speeding-up inverse optimization of boundary conditions for PDEs via backpropagation through time in latent space, and an annealing technique to address the non-differentiability and sparse interaction of boundary conditions. We test our method in a 1D benchmark of nonlinear PDEs, 2D Navier-Stokes flows into turbulent phase and an inverse optimization of boundary conditions in 2D Navier-Stokes flow. Compared to state-of-the-art deep learning-based surrogate models and other strong baselines, we demonstrate up to 128x reduction in the dimensions to update, and up to 15x improvement in speed, while achieving competitive accuracy.

Key facts. Cited by 39 papers in the map (data through 14 Oct 2026). Builds on 39 of its 94 references that are in the map. First posted on arXiv 15 Jun 2022.

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39 of 94 references.

References · 39

  1. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  2. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  3. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,713
  4. 2016SGDR: Stochastic Gradient Descent with Warm Restartscited by 5,045
  5. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  6. 2015Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networkscited by 2,054
  7. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  8. 2018Group Normalizationcited by 1,524
  9. 20163D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotationcited by 1,476
  10. 2018Learning Latent Dynamics for Planning from Pixelscited by 1,264
  11. 2015Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)cited by 1,070
  12. 2020Learning to Simulate Complex Physics with Graph Networkscited by 880
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  2. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  3. 2021Machine learning accelerated computational fluid dynamicscited by 471
  4. 2022Message Passing Neural PDE Solverscited by 386
  5. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  6. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 235
  7. 2019Stochastic Latent Actor-Critic: Deep Reinforcement Learning with a Latent Variable Modelcited by 231
  8. 2019DeepMDP: Learning Continuous Latent Space Models for Representation Learningcited by 207
  9. 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 205
  10. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 204
  11. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  12. 2019Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networkscited by 145
  13. 2020Learning to Control PDEs with Differentiable Physicscited by 132
  14. 2018Deep Fluids: A Generative Network for Parameterized Fluid Simulationscited by 121
  15. 2018Relational Neural Expectation Maximization: Unsupervised Discovery of Objects and their Interactionscited by 100
  16. 2018Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flowcited by 97
  17. 2017Visual Interaction Networkscited by 95
  18. 2020Multiscale Simulations of Complex Systems by Learning their Effective Dynamicscited by 86
  19. 2019A deep-learning-based surrogate model for data assimilation in dynamic subsurface flow problemscited by 59
  20. 2022Physical Design using Differentiable Learned Simulatorscited by 37
  21. 2022Learning to Solve PDE-constrained Inverse Problems with Graph Networkscited by 37
  22. 2020Implicit Rank-Minimizing Autoencodercited by 35
  23. 2020Symbolic Pregression: Discovering Physical Laws from Distorted Videocited by 21
  24. 2020Learning Physical Constraints with Neural Projectionscited by 19
  25. 2020Latent Space Subdivision: Stable and Controllable Time Predictions for Fluid Flowcited by 13
  26. 2020Phase Transitions for the Information Bottleneck in Representation Learningcited by 11
  27. 2014Galaxy Formation: Bayesian History Matching for the Observable Universecited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 3; 2023: 7; 2024: 13; 2025: 6; 2026: 10.

39 papers.

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  5. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulationcited by 71
  6. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 28
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  13. 2026Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solvingcited by 1
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  27. 2026Learn the Dynamics, Enforce the Physics: Generative Latent Operators for High-Dimensional Beam Evolutioncited by 0