A framework to solve inverse problems for parametric PDEs using adaptive finite elements and neural networks

Aug 6, 2026

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Parameter identification plays a central role in engineering workflows and often requires many repeated evaluations. This work introduces a reduction-based framework for inverse problems involving parametric partial differential equations. The reduced model employs a neural-network approximation trained on data produced by adaptive finite element simulations. A supervised feedforward network enables fast online evaluation of the parametric PDE solution. The inverse task identifies unknown parameters by minimizing a misfit functional using particle swarm optimization, with numerical results for elliptic and hyperbolic models.

LanguageEnglish

CategoryResearch & Report

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Introduction

Direct and inverse problem setting

Adaptive finite elements—neural networks method for the direct problem

Neural network approximation and training data generation

Alexandre Caboussat, Maude Girardin, and Marco Picasso A framework to solve inverse problems for parametric PDEs using adaptive finite elements and neural networks Abstract: Parameter identification is important in many engineering processes, and benefits from quick evaluations with reduced models. We present here a framework to solve inverse problems for parametric partial differential equations. The reduced model is built on a neural network method for the numerical approximation of a given parametric partial differential equation. Training data are generated thanks to adaptive finite element simulations. A supervised feedforward neural network is then used for the online approximation of the solution. The inverse problem aims at identifying parameters using available measurement data. The corresponding optimization problem is solved with a particle swarm optimization method. Numerical results arepresented for the parameter identification of several elliptic and hyperbolic model problems. Keywords: Finite element method, parametric PDEs, neural networks, adaptive mesh refinement, inverse problem, particle swarm optimization MSC 2020: 65N15, 65N30, 65N50, 68T07, 65M32 1 Introduction We consider parametric partial differential equations that are either stationary ℱ(u(x; μ); μ) = 0 x ∈ Ω, μ ∈ 􀁐 , (1.1) or evolutive ℱ(u(x, t; μ); μ) = 0 x ∈ Ω, t ∈ [0, T], μ ∈ 􀁐 , (1.2) Alexandre Caboussat, Geneva School of Business Administration (HEG-Genève), University of Applied Sciences and Arts Western Switzerland (HES-SO), 17 Rue de la Tambourine, 1227 Carouge, Geneva, Switzerland, e-mail: [alexandre.caboussat@hesge.ch](alexandre.caboussat@hesge.ch) Maude Girardin, Geneva School of Business Administration (HEG-Genève), University of Applied Sciences and Arts Western Switzerland (HES-SO), 17 Rue de la Tambourine, 1227 Carouge, Geneva, Switzerland; and Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, e-mail: [maude.girardin@epfl.ch](maude.girardin@epfl.ch) Marco Picasso, Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, [e-mail: marco.picasso@epfl.ch](e-mail: marco.picasso@epfl.ch) Open Access. © 2025 the author(s), published by Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution-NoDerivatives 4 .0 International License. [https://doi.org/10.1515/9783111376776-001](https://doi.org/10.1515/9783111376776-001) 2 􀱪 A. Caboussat et al. where Ω ⊆ ℝd (d ≥ 1) is the physical space, 􀁐 ⊂ ℝp is the parameter space (with p ≥ 1 possibly large), and ℱ is some differential operator. Given observed values of the solution ui at given (space or space-time) locations Yi, i = 1 . . . , Nobs, the goal is to determine the optimal parameter μ∗ ∈ 􀁐 that minimizes the objective function F(μ) := N1obs ibs1 􀀡 u(Yi; μ) − ui 􀀡 α , (1.3) where u(⋅; μ) is the solution of the parametric PDE for a given value of the parameter μ, and where α > 0 is given (typically α = 1 or α = 2) . We do not consider noise in the data ui, and thus do not investigate potential smoothing terms. Any optimization method to minimize F(μ) requires to compute (an approximation of) the solution u(⋅; μ) a large number of times. Such evaluation should thus be fast, hence the advantage to build and use a reduced model to approximate u. Depending on the type of PDE, several reducedorder modeling approaches may be considered, such as, e. g., reduced basis [10], proper orthogonal decomposition [24], polynomial chaos expansion [6], or neural networks [8, 11, 25] . Solving nonlinear optimization problems, including inverse problems, requires dedicated computational optimization methods. Among them, we can mention Newtontype approaches [7] or derivative-free approaches [1, 13, 18, 22] . We will in particular focus here on particle swarm optimization (PSO) [12, 16, 20]. On the other hand, among reduced-order models, we consid

FAQs

What is the main goal of the proposed framework?

It aims to solve inverse problems for parametric PDEs by identifying unknown parameters using available measurement data and fast reduced-order evaluations.

How is the reduced model constructed and trained?

A feedforward neural network approximates the parametric PDE solution. Training data are generated with adaptive finite element simulations to control accuracy across the parameter space.

How are the parameters identified from measurements?

An optimization problem is set up to minimize a misfit objective between neural-network-predicted solutions and observed data. Particle swarm optimization is used to solve this minimization.

Aug 6, 2026

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