# Physics-informed machine learning

@inproceedings{Wahlstrm2021PhysicsinformedML, title={Physics-informed machine learning}, author={Niklas Wahlstr{\"o}m and Adrian G. Wills and Johannes N. Hendriks and Alexander Gregg and Christopher M. Wensrich and A. Solin and Simo S{\"a}rkk{\"a}}, year={2021} }

| Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and highdimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning…

## 81 Citations

Characterizing possible failure modes in physics-informed neural networks

- Computer Science, MathematicsArXiv
- 2021

It is demonstrated that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena for even slightly more complex problems.

Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data

- Mathematics, Computer ScienceArXiv
- 2021

This paper proposes to use Bayesian neural networks (BNN) in order to recover the full system states from measurement data, and uses Hamiltonian Monte-Carlo to sample the posterior distribution of a deep and dense BNN, showing that it is possible to accurately capture physics of varying complexity, without overfitting.

Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations

- Physics, Computer ScienceArXiv
- 2021

Numerical experiments show that FBPINNs are effective in solving both small and larger, multi-scale problems, outperforming standard PINNs in both accuracy and computational resources required, potentially paving the way to the application of PINNs on large, real-world problems.

Interpolating between BSDEs and PINNs - deep learning for elliptic and parabolic boundary value problems

- Computer Science, MathematicsArXiv
- 2021

This paper reviews the literature and suggests a methodology based on the novel diffusion loss that interpolates between BSDEs and PINNs, which opens the door towards a unified understanding of numerical approaches for high-dimensional PDEs, as well as for implementations that combine the strengths of BS DEs andPINNs.

Optimal control of PDEs using physics-informed neural networks

- Mathematics, Physics
- 2021

Physics-informed neural networks (PINNs) have recently become a popular method for solving forward and inverse problems governed by partial differential equations (PDEs). By incorporating the…

Neural Networks with Inputs Based on Domain of Dependence and A Converging Sequence for Solving Conservation Laws, Part I: 1D Riemann Problems

- Computer Science, MathematicsArXiv
- 2021

2-Coarse-Grid neural network and 2-Diffusion-Coefficient neural network are introduced, which use 2 solutions of a conservation law from a converging sequence, computed from a low-cost numerical scheme, and in a domain of dependence of a space-time grid point as the input for a neural network to predict its high-fidelity solution at the grid point.

PHYSICS-INFORMED RANDOM PROJECTION NEURAL NETWORKS

- 2021

We propose a numerical method based on physics-informed Random Projection Neural Networks for the solution of Initial Value Problems (IVPs) of Ordinary Differential Equations (ODEs) with a focus on…

Physics and Equality Constrained Artificial Neural Networks: Application to Partial Differential Equations

- Computer Science, PhysicsArXiv
- 2021

The efficacy and versatility of the physics- and equality-constrained deep-learning framework are demonstrated by applying it to learn the solutions of various multi-dimensional PDEs, including a nonlinear inverse problem from the hydrology field with multi-fidelity data fusion.

A hybrid physics-informed neural network for nonlinear partial differential equation

- Computer Science, MathematicsArXiv
- 2021

This paper focuses on the discrete time physics-informed neural network (PINN), and proposes a hybrid PINN scheme for the nonlinear PDEs, which has a better performance in approximating the discontinuous solution even at a relatively larger time step.

Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training

- Computer ScienceArXiv
- 2021

This work presents a PINN approach to solving the equations of coupled flow and deformation in porous media for both single-phase and multiphase flow, and proposes a sequential training approach based on the stress-split algorithms of poromechanics.

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