Physics-Informed Machine Learning

Joint Bayesian Inference on Lagrangian Physics and Trajectories

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Abstract

Numerical integration and ODE discovery are two sides of the same coin—converse problems of finding trajectories from known physics versus inferring physics from observed trajectories. Although these problems have been extensively studied in isolation, they can be unified through the minimization of a common quantity: the Euler–Lagrange residual. In this paper, we build on this insight and introduce the Integrated Squared Action Residual (ISAR), which enables both tasks to be performed simultaneously. We formulate numerical integration and model discovery as a joint Bayesian inference problem, allowing for the systematic incorporation of physical prior knowledge and domain constraints in settings with sparse and noisy observations, where traditional approaches often fail. While we demonstrate the performance on two mechanical toy problems, it can be readily extended towards multiphysics systems including dissipative dynamics.

How to Cite:

Obermayr, M. & Peharz, R., (2026) “Joint Bayesian Inference on Lagrangian Physics and Trajectories”, Proceedings of the Austrian Symposium on AI, Robotics, and Vision 3(1), 122-127.