Maximum likelihood discretization of the transport equation

2025/05/15 by Brook Eyob, Florian Schäfer, Eyob, Brook +1
Mathematics · Physics and Astronomy · #35L65 #58B20 #65M25 #76J20 #76L05 #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Fractional Differential Equations Solutions #Markov Chains and Monte Carlo Methods #Numerical Analysis (math.NA) #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2505.10713

openalex publication_date 2025/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The transport of positive quantities underlies countless physical processes, including fluid, gas, and plasma dynamics. Discretizing the associated partial differential equations with Galerkin methods can result in spurious nonpositivity of solutions. We observe that these methods amount to performing statistical inference using the method of moments (MoM) and that the loss of positivity arises from MoM's susceptibility to producing estimates inconsistent with the observed data. We overcome this problem by replacing MoM with maximum likelihood estimation, introducing maximum likelihood discretization (MLD). In the continuous limit, MLD simplifies to the Fisher-Rao Galerkin (FRG) semidiscretization, which replaces the L² inner product in Galerkin projection with the Fisher-Rao metric of probability distributions. We show empirically that FRG preserves positivity. We prove rigorously that it yields error bounds in the Kullback-Leibler divergence.

Citations

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