Adjoint-based gradient estimation from gray-box solutions of unknown conservation laws

TL;DR

This paper presents a novel approach to estimate gradients in gray-box conservation law simulators by inferring unknown flux functions from solutions and then applying the adjoint method, enabling gradient-based optimization.

Contribution

It introduces a method to infer flux functions from solutions in gray-box simulators and compute gradients via the adjoint method, addressing a key limitation.

Findings

Effective gradient estimation demonstrated on flow problems

Enables optimization in gray-box conservation law models

Addresses unknown flux function challenge

Abstract

Many engineering applications can be formulated as optimizations constrained by conservation laws. Such optimizations can be efficiently solved by the adjoint method, which computes the gradient of the objective to the design variables. Traditionally, the adjoint method has not been able to be implemented in many "gray-box" conservation law simulators. In gray-box simulators, the analytical and numerical form of the conservation law is unknown, but the full solution of relevant flow quantities is available. In this paper, we consider the case where the flux function is unknown. This article introduces a method to estimate the gradient by inferring the flux function from the solution, and then solving the adjoint equation of the inferred conservation law. This method is demonstrated in the sensitivity analysis of two flow problems.