DeepAI AI Chat
Log In Sign Up

Optimizing a DIscrete Loss (ODIL) to solve forward and inverse problems for partial differential equations using machine learning tools

by   Petr Karnakov, et al.

We introduce the Optimizing a Discrete Loss (ODIL) framework for the numerical solution of Partial Differential Equations (PDE) using machine learning tools. The framework formulates numerical methods as a minimization of discrete residuals that are solved using gradient descent and Newton's methods. We demonstrate the value of this approach on equations that may have missing parameters or where no sufficient data is available to form a well-posed initial-value problem. The framework is presented for mesh based discretizations of PDEs and inherits their accuracy, convergence, and conservation properties. It preserves the sparsity of the solutions and is readily applicable to inverse and ill-posed problems. It is applied to PDE-constrained optimization, optical flow, system identification, and data assimilation using gradient descent algorithms including those often deployed in machine learning. We compare ODIL with related approach that represents the solution with neural networks. We compare the two methodologies and demonstrate advantages of ODIL that include significantly higher convergence rates and several orders of magnitude lower computational cost. We evaluate the method on various linear and nonlinear partial differential equations including the Navier-Stokes equations for flow reconstruction problems.


page 1

page 2

page 3

page 4


Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory

Deep learning has significantly revolutionized the design of numerical a...

Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces

We introduce a novel spectral, finite-dimensional approximation of gener...

NTopo: Mesh-free Topology Optimization using Implicit Neural Representations

Recent advances in implicit neural representations show great promise wh...

ViTO: Vision Transformer-Operator

We combine vision transformers with operator learning to solve diverse i...

A Forward Propagation Algorithm for Online Optimization of Nonlinear Stochastic Differential Equations

Optimizing over the stationary distribution of stochastic differential e...

Accelerating numerical methods by gradient-based meta-solving

In science and engineering applications, it is often required to solve s...

Competitive Physics Informed Networks

Physics Informed Neural Networks (PINNs) solve partial differential equa...