Resumo: Parameter estimation in differential equation models is a fundamental challenge in engineering applications requiring accurate estimation of physical parameters for system design and optimization. This paper presents a Bayesian framework for solving inverse problems in differential equation-based models. Two approaches are developed: a mesh-based nonlinear regression method and a mesh-free approach based on Bayesian physics-informed Gaussian processes that eliminates mesh dependency while incorporating physical constraints through differential linear operators. Hamiltonian Monte Carlo and automatic differentiation variational inference are implemented and compared for posterior approximation. The framework is demonstrated through three applications: advection-diffusion-reaction systems for environmental contamination modelling, exponential growth models for thermal property estimation, and the nonlinear Burgers equation via the Cole-Hopf transformation.
Short bio: Sou professor do Departamento de Métodos Estatísticos e do Programa de Pós Graduação em Estatística da UFRJ desde 2020. Obtive doutorado em Estatística e Ciência de Dados pela University of Texas at Austin (2019), mestrado em Estatística pela Universidade Federal do Rio de Janeiro (2015) e bacharelado em Estatística também pela Universidade Federal do Rio de Janeiro (2014) e . Atualmente, tenho trabalhado nas áreas de Bioestatística, Inferência Bayesiana, Estatística Espacial e Redes Neurais.
