References

This page collects key references for AI methods in physics simulation, with a focus on constrained mechanics and differentiable solvers.

Graph Neural Networks for Simulation

• MeshGraphNets: Learning Mesh-Based Simulation with Graph Networks
  https://arxiv.org/abs/2010.03409

• ConstraintGNS: Constraint-based Graph Network Simulator
  https://proceedings.mlr.press/v162/rubanova22a/rubanova22a.pdf

• MPNPDE: Message Passing Neural PDE Solvers
  https://openreview.net/forum?id=vSix3HPYKSU

Equivariant Architectures

• TFN: Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds https://arxiv.org/abs/1802.08219

• SE3Transformer: SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks https://proceedings.neurips.cc/paper/2020/hash/15231a7ce4ba789d13b722cc5c955834-Abstract.html

• Hamiltonian Neural Networks: Hamiltonian Neural Networks
  Sam Greydanus, Misko Dzamba, Jason Yosinski
  https://arxiv.org/abs/1906.01563

• Port-Hamiltonian Structure of Continuum Mechanics
  Ramy Rashad & Stefano Stramigioli
  Journal of Nonlinear Science, Volume 35, Article 35 (2025).
  Published: 28 January 2025
  https://link.springer.com/article/10.1007/s00332-025-10130-1

• Graph Attention Hamiltonian Neural Networks: A Lattice System Analysis Model Based on Structural Learning
  Ru Geng*, Yixian Gao*, Jian Zu**, Hong-Kun Zhang**
  Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University; University of Massachusetts Amherst; Great Bay University
  https://arxiv.org/abs/2412.10821

Neural Operators

• DeepONet: Learning nonlinear operators via DeepONet
  https://www.nature.com/articles/s42256-021-00302-5

• FNO: Fourier Neural Operator
  https://arxiv.org/abs/2010.08895

• PINO: Physics-Informed Neural Operator (PINO)
  https://arxiv.org/abs/2111.03794

• GKN: Neural Operator: Graph Kernel Network for PDEs
  https://openreview.net/forum?id=fg2ZFmXFO3

• GeoFNO: Fourier Neural Operator with Learned Deformations (Geo-FNO)
  https://jmlr.org/papers/v24/23-0064.html

• RIGNO: RIGNO: Robust operator learning on arbitrary domains
  https://arxiv.org/abs/2501.19205

• UniversalNeuralOperators: Towards Universal Neural Operators through Multiphysics Pretraining
  https://arxiv.org/abs/2511.10829

Implicit Models and DEQ

• DEQ: Deep Equilibrium Models
  https://arxiv.org/abs/1909.01377

• IGNN: Implicit Graph Neural Networks
  https://arxiv.org/abs/2009.06211

• ContinuousDEQ: Mixing Implicit and Explicit Deep Learning with Skip DEQs and Infinite Time Neural ODEs
  https://arxiv.org/abs/2201.12240

• IGNNSolver: IGNN-Solver
  https://arxiv.org/abs/2410.08524

• ImplicitVsUnfolded: Implicit vs Unfolded Graph Neural Networks
  https://www.jmlr.org/papers/v26/22-0459.html

Solver Assistance

• LearnedCGPrecond: Learning Preconditioners for Conjugate Gradient PDE Solvers
  https://proceedings.mlr.press/v202/li23e.html

• NeuralKrylov: Neural Krylov Iteration
  https://proceedings.neurips.cc/paper_files/paper/2024/hash/e88870ec82f2469b0ddf32c817920c68-Abstract-Conference.html

• GNNPrecond: Graph Neural Preconditioners for Sparse Linear Systems
  https://openreview.net/forum?id=Tkkrm3pA35

Learned Constitutive Models

• NeuralConstitutive: Neural Constitutive Modeling of Soft Materials
  https://www.tandfonline.com/doi/full/10.1080/15376494.2024.2439557

• DEQuifyForceField: DEQuify your force field: More efficient simulations using deep equilibrium models
  https://openreview.net/forum?id=rynb4Vn8rb

Physics-Informed Methods

• BPINNs: B-PINNs (Bayesian PINNs)
  https://arxiv.org/abs/2003.06097

• PhysicsInformedOperators: Physics-Informed Neural Networks and Neural Operators for Parametric PDEs: A Human-AI Collaborative Analysis
  https://arxiv.org/html/2511.04576v1

• HemodynamicsPINN: Hemodynamics modeling with physics-informed neural networks
  https://www.sciencedirect.com/science/article/abs/pii/S0045782525001239

• EquivariantPINNs: Equivariant Neural Networks and Differential Invariants Theory for Solving Partial Differential Equations
  https://www.mdpi.com/2673-9984/5/1/13

Neural ODE and Continuous-Time Surrogates

• DEQNeuralODE: Acceleration of Dynamic Simulations using a Deep Equilibrium Layer and Neural ODE Surrogate
  https://arxiv.org/abs/2405.06827

• LiquidTimeConstant: Liquid Time-constant Networks
  https://arxiv.org/abs/2006.04439

• CfC: Closed-form Continuous-time Neural Networks
  https://arxiv.org/abs/2106.13898

Reduced-Order and Latent Dynamics

• LatentNeuralPDESolver: Latent Neural PDE Solver
  https://arxiv.org/abs/2402.17853

• KoopmanDL: Deep learning for Koopman linear embeddings
  https://www.nature.com/articles/s41467-018-07210-0

• RO-HNN: Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach https://arxiv.org/html/2509.24627v1

• SymplecticEncoders: Symplectic encoders for physics-constrained variational dynamics inference https://www.nature.com/articles/s41598-023-29186-8

Constrained Mechanics and SOFA

• SOFA Constrained Mechanics: Constrained mechanics formulation for deformable objects
  https://inria.hal.science/hal-00681539

• SOFA Numerical Resolution: Numerical resolution of constrained dynamics
  https://inria.hal.science/hal-01649355

JAX-Based Differentiable Physics Frameworks

• jaxdf: Differentiable Projective Dynamics (JAX-based differentiable physics simulation)
  Framework for differentiable physics simulation with constraints, enabling optimization and control of mechanical systems.
  https://github.com/ucl-bug/jaxdf

• JAX-FEM: Finite Element Method in JAX
  JAX-based FEM library enabling automatic differentiation of finite element simulations, useful for shape optimization and learned physics models.
  https://github.com/deepmodeling/jax-fem

These frameworks leverage JAX's automatic differentiation and JIT compilation capabilities to provide efficient, differentiable simulation tools for mechanics and physics applications.