# Learning to Control PDEs with Differentiable Physics

arXiv: https://arxiv.org/abs/2001.07457 · HTML: https://mlsanity.com/paper/2001.07457/

## Key facts

- Cited by 132 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jan 2020.

## Built on by (most cited first, 30 of 132)

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- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [DiffPD: Differentiable Projective Dynamics](https://mlsanity.com/paper/2101.05917/) (2021)
- [Fast and Feature-Complete Differentiable Physics for Articulated Rigid Bodies with Contact](https://mlsanity.com/paper/2103.16021/) (2021)
- [NeuPhysics: Editable Neural Geometry and Physics from Monocular Videos](https://mlsanity.com/paper/2210.12352/) (2022)
- [MotionCraft: Physics-based Zero-Shot Video Generation](https://mlsanity.com/paper/2405.13557/) (2024)
- [ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2408.11104/) (2024)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [NTopo: Mesh-free Topology Optimization using Implicit Neural Representations](https://mlsanity.com/paper/2102.10782/) (2021)
- [FluidLab: A Differentiable Environment for Benchmarking Complex Fluid Manipulation](https://mlsanity.com/paper/2303.02346/) (2023)
- [Efficient Differentiable Simulation of Articulated Bodies](https://mlsanity.com/paper/2109.07719/) (2021)
- [Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons](https://mlsanity.com/paper/2202.06988/) (2022)
- [Differentiable Simulation of Soft Multi-body Systems](https://mlsanity.com/paper/2205.01758/) (2022)
- [Learning to Accelerate Partial Differential Equations via Latent Global Evolution](https://mlsanity.com/paper/2206.07681/) (2022)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [Learning to Solve PDE-constrained Inverse Problems with Graph Networks](https://mlsanity.com/paper/2206.00711/) (2022)
- [NeuroFluid: Fluid Dynamics Grounding with Particle-Driven Neural Radiance Fields](https://mlsanity.com/paper/2203.01762/) (2022)
- [A posteriori learning for quasi-geostrophic turbulence parametrization](https://mlsanity.com/paper/2204.03911/) (2022)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Learning neural network potentials from experimental data via Differentiable Trajectory Reweighting](https://mlsanity.com/paper/2106.01138/) (2021)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Causal Navigation by Continuous-time Neural Networks](https://mlsanity.com/paper/2106.08314/) (2021)

## Abstract (arXiv)

Predicting outcomes and planning interactions with the physical world are long-standing goals for machine learning. A variety of such tasks involves continuous physical systems, which can be described by partial differential equations (PDEs) with many degrees of freedom. Existing methods that aim to control the dynamics of such systems are typically limited to relatively short time frames or a small number of interaction parameters. We present a novel hierarchical predictor-corrector scheme which enables neural networks to learn to understand and control complex nonlinear physical systems over long time frames. We propose to split the problem into two distinct tasks: planning and control. To this end, we introduce a predictor network that plans optimal trajectories and a control network that infers the corresponding control parameters. Both stages are trained end-to-end using a differentiable PDE solver. We demonstrate that our method successfully develops an understanding of complex physical systems and learns to control them for tasks involving PDEs such as the incompressible Navier-Stokes equations.