• PDE applications
  • Neural Operators (on function space)
  • PDE Solvers (ivp/bvp)
    • Physics-informed neural networks
    • Linear regression
• Physical Systems
  • Spatiotemporal Dynamics
  • Kinetics
• Differential equation enabled neural networks

PDE applications

Neural Operators (on function space)

• DeepONet

  branch net for input function (with fixed input location) and Trunk net at target locations paper

• Neural Operators (link)

  • Graph kernel network

    paper: uniform mesh (graph message passing), ‘decompose’ a target spatial function

  • Multipole graph kernel network

    paper (NIPS 2020): hierarchical graph kernels

  • Fourier neural operator

    paper (ICLR 2021): fourier decomposition

  • Geo-FNO

    paper: FNO on arbitrary geometries

  • Markov neural operator

    paper (NIPS 2022): sequential and dissipative systems; learning global attractor

PDE Solvers (ivp/bvp)

Physics-informed neural networks

• PINN

  paper: define loss to specify constraints

Linear regression

• PDE solver with convergence guarentees

  original paper (ICLR 2019)

  • PDE class: Poisson (laplacian) $\nabla^2 u = f$
    • elements: linear $Au = f$, f, boundary B, boundary condition b, discretization n
    • fixed: A
    • vary: f, B, b, n

• Multigrid PDE solver

  original paper

  • PDE class: diffusion $\nabla (\mathbf{g}\nabla \mathbf{u}) = \mathbf{f}$
    • elements: linear $Au=f$, f, boundary B, boundary condition b, discretization n
    • vary: b, n, f(?)

• Message-passing neural PDE (MP-PDE)

  original paper (ICLR 2022)

Physical Systems

Spatiotemporal Dynamics

• Kinetics

• Hamiltonian Neural Networks (HNN)

  Hamiltonian equation, paper (NIPS 2019)

  deconstructing the inductive biases of HNN, paper (ICLR 2022)

• Symplectic RNN (SRNN)

  Hamiltonian Net with RNN as ODE solver, paper (ICLR 2020)

• Deep Lagrangian Network (DeLaN)

  Lagrangian mechanical system, paper (ICLR 2019)

• Lagrangian neural networks

  general coordinates, paper

Differential equation enabled neural networks

• Neural ODE (NODE)

  neural function as the differential equation, paper (NIPS 2018)

• Augmented NODEs

  augment space dimensions to improve expressiveness, paper (NIPS 2019)

• Dissecting NODEs

  Continuous and variance depth parameter model, encoder/decoder (augmentation), data-control and depth-adaptation paper (NIPS 2020)

• Second-order NODEs (SONODEs)

  A seperate function to learn the second order time derivative paper (NIPS 2020)

• Characteristic NODE (C-NODE)

• Graph NODEs (GDEs)

  GNN + NODE