Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification - paper

Aug 5, 2026

Machine learning identifies discrete variational principles for field theories from observed solutions using Gaussian process regression to infer a discrete Lagrangian density. The approach is geometric: the learned data-driven model reproduces the variational structure by design. A rigorous convergence result is provided, addressing ambiguity in discrete Lagrangian densities within inverse variational calculus. The method also enables uncertainty quantification for model predictions, including equations of motion and linear observables, demonstrated on discrete wave and Schrödinger equations.

LanguageEnglish

CategoryResearch & Report

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Pages28

Reading Time1 h 11 min

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Introduction

Continuous Lagrangian data-driven models

Discrete Lagrangian data-driven models

arXiv :2407 .07642v1 [math .NA] 10 Jul 2024 Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification Christian Offen Paderborn University, Department of Mathematics Warburger Str. 100, 33098 Paderborn, Germany christian.offen@uni-paderborn.de July 11, 2024 We introduce a method based on Gaussian process regression to identify discrete variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a geometric machine learning technique in the sense that the variational structure of the true field theory is reflected in the datadriven model by design. We provide a rigorous convergence statement of the method. The proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus. Moreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory. This is illustrated on the example of the discrete wave equation and Schr¨odinger equation. The article constitutes an extension of our previous article for the data-driven identification of (discrete) Lagrangians for variational dynamics from an ode setting to the setting of discrete pdes. 1. Introduction Partial differential equations that can be derived from variational principles (field theories) take a prominent role in physics, molecular biology, and engineering, for instance, as they describe wave phenomena, evolution of plasma dynamics, electro-magnetism, fluid dynamics, and quantum mechanical processes. Examples include the Korteweg–De Vries (KdV) equation, the shallow wave equation, and the Schr¨odinger equation. In this context, data-driven technology has been used to obtain solutions to field theories [13] as well as to discover governing equations from observed solutions [29, 26] . This article falls into the latter category. We propose to identify an action functional for afield theory by identifying a discrete Lagrangian density based on Gaussian process regression. We do not assume any prior knowledge of the specific form of the Lagrangian. Central novelty of the article is the provision of a rigorous convergence theory for the proposed method. Moreover, Lagrangian descriptions of field theories are highly ambiguous [12, 17, 16] . This has practical implications for our machine learning framework that we discuss systematically and account for in the convergence theory. Furthermore, we provide systematic uncertainty quantification of linear observables of the data-driven system. The article can be seen as an extension of our article [24] from the context of ordinary differential equations to a partial differential equations’ context. Continuous Lagrangian data-driven models A dynamical system is governed by avariational principle or a least action principle, if motions constitute critical points of an action functional. In case of an autonomous first-order field theory in the space of Rn, the action functional is of the form S (u) = Z L (u(t), ut1 ,..., utn)dt1 ... dtn , (1) where u is a scalar field and ut k = ~~u~~t~~ ~~k denote its derivatives (k = 1 ,..., n) . The function L is referred to as a Lagrangian density of the field theory. In the special case of n = 1 the function L is called Lagrangian. A function u: Rn → R is a solution of the field theory if for any bounded, open domain M ⊂ Rn with Lebesgue measurable boundary ∂M the action S is stationary at u|M for all variations δu: M → R that fix ∂M. This is (under regularity assumptions) equivalent to the condition that u fulfils the Euler-Lagrange equations EL(u) = 0, where EL(u) = ∂∂tk 􀀒 tk~~ ~~ 􀀓 − Lu = kX, 􀀒 ∂u∂t2kut~~ ~~l~~ ~~ utk,tl􀀓 − Lu . (2) Here, ut k,tl = ∂~~∂~~t2ktl . Details may be found in [11, 32], for instance. In the data-driven context, L is sought as a function of u and its gradien

FAQs

How does the method learn variational principles from data?

It uses Gaussian process regression to identify a discrete Lagrangian density directly from observed solutions at discrete spacetime locations.

What does the convergence statement guarantee?

The article provides a rigorous convergence theory for the proposed identification method despite ambiguity of discrete Lagrangian densities in inverse variational calculus.

How is uncertainty quantified in the learned discrete field theory?

The framework quantifies model uncertainty for the equations of motion and for any linear observable defined on the discrete field theory.

Aug 5, 2026

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