Deep Conservation: A latent-dynamics model for exact satisfaction of physical conservation laws

TL;DR

This paper introduces a deep learning approach that learns low-dimensional representations of dynamical systems while exactly enforcing physical conservation laws through constrained optimization in the latent space.

Contribution

It presents a novel latent-dynamics model that guarantees exact satisfaction of conservation laws by integrating them as explicit constraints in the learning process.

Findings

Ensures conservation laws are exactly satisfied in the learned dynamics.

Uses deep autoencoders to find low-dimensional embeddings of high-dimensional systems.

Provides a framework for physically consistent reduced-order modeling.

Abstract

This work proposes an approach for latent-dynamics learning that exactly enforces physical conservation laws. The method comprises two steps. First, the method computes a low-dimensional embedding of the high-dimensional dynamical-system state using deep convolutional autoencoders. This defines a low-dimensional nonlinear manifold on which the state is subsequently enforced to evolve. Second, the method defines a latent-dynamics model that associates with the solution to a constrained optimization problem. Here, the objective function is defined as the sum of squares of conservation-law violations over control volumes within a finite-volume discretization of the problem; nonlinear equality constraints explicitly enforce conservation over prescribed subdomains of the problem. Under modest conditions, the resulting dynamics model guarantees that the time-evolution of the latent state exactly satisfies conservation laws over the prescribed subdomains.