NeuralPDE.jl icon indicating copy to clipboard operation
NeuralPDE.jl copied to clipboard

Restructuring the NeuralPDE package and its docs

Open finmod opened this issue 3 years ago • 0 comments

Introduce in the "features" section of docs, a roadmap of development of NeuralPDE along the line of section 3 of https://arxiv.org/abs/2101.08068 that is in two parts: A deterministic approach by NN for nonlinear 1st order PDEs and A probabilistic approach by NN for nonlinear 2nd order PDEs.

Generalize ModelingToolkit formulation of all PDE problems and examples with common/standard dictionary of notation and operators: Nabla, Inverse Nabla, div, Expectation, Dynamic Programming Principle, Stochastic DDP, Euler-Lagrange ODE system, Legendre transform as in http://math.stanford.edu/~ryzhik/STANFORD/MEAN-FIELD-GAMES/notes-mean-field.pdf. This leads to classify the types of PDE problems : Initial/Terminal conditions, boundary conditions, boundary free problems, variational inequalities, high dimensional systems of same type, forward-backward systems. Here, one should note if the PDE problem is solved by optimize-discretize (=> NeuralPDE) or discretize-optimize (=> DiffEqOperators).

At the outset of the NeuralPDE docs, a synopsis table of which algorithm should be used/available for each type of problem. For the deterministic approach: Deep Galerkin Method, PINNs, Deep Ritz method and what is currently in NeuralPDE. The same for the probabilistic approach: Deep BSDE schemes, Deep Backward Dynamic Programming.

The specific API documentation should a give a pseudocode of what is going on and where the NN are applied as in https://maziarraissi.github.io/research/1_physics_informed_neural_networks/ . Otherwise, the technicalities of the NN overwhelm the concept of where and how the NN is applied.

Wrap open issues #20, #52, #112 and possibly others in this one and close them.

finmod avatar Jun 16 '21 13:06 finmod