ScimBa

This librairies impliment varying SciML methods for varying PDE problem and also tools to build hybrid numerical methods.

Current Content

• Nets: MLP networks, Discontinuous MLP, RBF networks, some activations functions and a basic trainer
• Sampling and domain: general uniform sampling methods for PINNs and Neural Operators. Sampling based on approximated signed distance function for general geometries.
• PDEs: the librairiy implement différent type of model: ODE, spatial pde, times-apce pde, stationary kinetic PDE and kinetic PDE.
• Specific networks for PINNs: For all the equations we implement PINNs networks based on: MLP, Discontinuous MLP and nonlinear radial basis function. We implement also the Fourier network with general features (Fourier but also Gaussian etc)
• Trainer: for each type of PDE we gives a specific trainer.
• Generative Nets: Normalized flows, Gaussian mixture. The classical and conditional approaches are proposed. Trainer based on the maximum likelihood principle.
• Neural Operator: Continuous Auto-encoder based on PointNet encoder and coordinate based decoder. Physic informed DeepOnet for ODE, spatial and time space PDE.
• Neural Galerkin: Method Neural Galerkin for time PDE based on the same network than PINNs.

Ongoing work for 2024

• Nets: New activation function used for implicit representation, Symbolic models, Sindy
• Sampling and domain: learning of signed distance function using PINNs, adaptive sampling
• Specific networks for PINNs: Multiscale architecture, spectral architecture for kinetic, specific architecture.
• Trainer: Trainer with sparsity constraints and globalization method. Loss Balancing
• Generative Nets: Energy models, score matching, more complex normalized flow, Continuous VAE
• Neural Operator: physic informed DeepGreen operator, FNO, GINO based on FNO, NO with neural implicit representation. Kinetic case
• Neural Galerkin: Adaptive sampling, randomization, Least Square solver, implicit scheme. CROM Space time reduced Galerkin model. Greedy basis.

References

PINNs and MLP

• https://www.sciencedirect.com/science/article/abs/pii/S0021999118307125
• https://arxiv.org/abs/2209.03984
• https://arxiv.org/abs/1912.00873
• https://arxiv.org/abs/1912.00873
• https://arxiv.org/abs/2109.01050
• https://arxiv.org/abs/2203.01360
• https://arxiv.org/abs/2103.09959
• https://openreview.net/forum?id=vsMyHUq_C1c

Neural Galerkin

• https://arxiv.org/abs/2306.15630
• https://arxiv.org/abs/2306.03749
• https://arxiv.org/abs/2207.13828
• https://arxiv.org/abs/2201.07953
• https://arxiv.org/abs/2104.13515

DeepOnet

• https://www.nature.com/articles/s42256-021-00302-5
• https://www.science.org/doi/10.1126/sciadv.abi8605
• https://arxiv.org/abs/2205.11404
• https://arxiv.org/abs/2206.03551

FNO and diverse geometry

• https://openreview.net/forum?id=c8P9NQVtmnO
• https://arxiv.org/abs/2207.05209
• https://arxiv.org/abs/2212.04689
• https://arxiv.org/abs/2305.00478
• https://arxiv.org/abs/2306.05697
• https://arxiv.org/abs/2305.19663

Other NO

• https://openreview.net/forum?id=LZDiWaC9CGL
• https://arxiv.org/abs/2205.10573
• https://arxiv.org/abs/2205.02191
• https://openreview.net/forum?id=kIo_C6QmMOM
• https://arxiv.org/abs/2303.10528
• https://arxiv.org/abs/2302.05925