Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations

TL;DR

This paper introduces a novel method combining moment and SDP relaxation techniques with maximum entropy estimation to produce smooth approximations of solutions to nonlinear differential equations, including ODEs, PDEs, and OCPs.

Contribution

It presents a new approach that integrates moment-based SDP relaxations with maximum entropy methods for smooth solution approximation of complex differential equations.

Findings

Effective for linear and nonlinear ODEs and PDEs

Provides smooth closed-form solutions from discrete approximations

Preliminary numerical results demonstrate potential

Abstract

Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of problems involving nonlinear differential equations. Given a system of nonlinear differential equations, we apply a technique based on finite differences and sparse SDP relaxations for polynomial optimization problems (POP) to obtain a discrete approximation of its solution. In a second step we apply maximum entropy estimation (using moments of a Borel measure associated with the discrete solution) to obtain a smooth closed-form approximation. The approach is illustrated on a variety of linear and nonlinear ordinary differential equations (ODE), partial differential equations (PDE) and optimal control problems (OCP), and preliminary numerical results are reported.