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Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers

arXiv 2007.00016 ↗First posted 30 Jun 2020physics.comp-ph cs.LGcited by 201

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Abstract (arXiv)

Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting for effects not captured by the discretized PDE. We target the problem of reducing numerical errors of iterative PDE solvers and compare different learning approaches for finding complex correction functions. We find that previously used learning approaches are significantly outperformed by methods that integrate the solver into the training loop and thereby allow the model to interact with the PDE during training. This provides the model with realistic input distributions that take previous corrections into account, yielding improvements in accuracy with stable rollouts of several hundred recurrent evaluation steps and surpassing even tailored supervised variants. We highlight the performance of the differentiable physics networks for a wide variety of PDEs, from non-linear advection-diffusion systems to three-dimensional Navier-Stokes flows.

Key facts. Cited by 201 papers in the map (data through 7 Oct 2026). First posted on arXiv 30 Jun 2020.

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Citing papers per year: 2021: 23; 2022: 49; 2023: 35; 2024: 27; 2025: 28; 2026: 39.

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  1. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  2. 2021Machine learning accelerated computational fluid dynamicscited by 461
  3. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  4. 2021Factorized Fourier Neural Operatorscited by 237
  5. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
  6. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  7. 2022Forecasting Global Weather with Graph Neural Networkscited by 165
  8. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstructioncited by 150
  9. 2023Scalable Transformer for PDE Surrogate Modelingcited by 123
  10. 2022Clifford Neural Layers for PDE Modelingcited by 91
  11. 2021Learned Coarse Models for Efficient Turbulence Simulationcited by 90
  12. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 90
38 more
  1. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  2. 2022Predicting Physics in Mesh-reduced Space with Temporal Attentioncited by 82
  3. 2022AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutionscited by 77
  4. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 76
  5. 2023DeepZero: Scaling up Zeroth-Order Optimization for Deep Model Trainingcited by 76
  6. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulationcited by 65
  7. 2022When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learningcited by 64
  8. 2021Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modelingcited by 59
  9. 2023Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamicscited by 56
  10. 2022Blending Neural Operators and Relaxation Methods in PDE Numerical Solverscited by 53
  11. 2022Physics Informed Neural Fields for Smoke Reconstruction with Sparse Datacited by 52
  12. 2022NeuPhysics: Editable Neural Geometry and Physics from Monocular Videoscited by 50
  13. 2022Learning to correct spectral methods for simulating turbulent flowscited by 49
  14. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 42
  15. 2021NTopo: Mesh-free Topology Optimization using Implicit Neural Representationscited by 41
  16. 2022Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizonscited by 39
  17. 2022Differentiable Simulation of Soft Multi-body Systemscited by 39
  18. 2021Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNscited by 38
  19. 2022Learning to Accelerate Partial Differential Equations via Latent Global Evolutioncited by 38
  20. 2022Learning to Solve PDE-constrained Inverse Problems with Graph Networkscited by 37
  21. 2022JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flowscited by 36
  22. 2022A posteriori learning for quasi-geostrophic turbulence parametrizationcited by 36
  23. 2022Learning differentiable solvers for systems with hard constraintscited by 34
  24. 2022Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamicscited by 30
  25. 2024DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systemscited by 29
  26. 2025PDE-Transformer: Efficient and Versatile Transformers for Physics Simulationscited by 29
  27. 2021Variational Data Assimilation with a Learned Inverse Observation Operatorcited by 28
  28. 2023Physics-Informed Machine Learning for Modeling and Control of Dynamical Systemscited by 28
  29. 2023NVFi: Neural Velocity Fields for 3D Physics Learning from Dynamic Videoscited by 28
  30. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  31. 2025Walrus: A Cross-Domain Foundation Model for Continuum Dynamicscited by 26
  32. 2023A Neural PDE Solver with Temporal Stencil Modelingcited by 22
  33. 2022Differentiable Programming for Earth System Modelingcited by 21
  34. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  35. 2023Learning Controllable Adaptive Simulation for Multi-resolution Physicscited by 20
  36. 2021Global Transport for Fluid Reconstruction with Learned Self-Supervisioncited by 18
  37. 2021Known Operator Learning and Hybrid Machine Learning in Medical Imaging -- A Review of the Past, the Present, and the Futurecited by 18
  38. 2024DiffPhyCon: A Generative Approach to Control Complex Physical Systemscited by 17

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