{"ID":2830762,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.09754","arxiv_id":"2512.09754","title":"On Parameter Identification in Three-Dimensional Elasticity and Discretisation with Physics-Informed Neural Networks","abstract":"Physics-informed neural networks have emerged as a powerful tool in the scientific machine learning community, with applications to both forward and inverse problems. While they have shown considerable empirical success, significant challenges remain -- particularly regarding training stability and the lack of rigorous theoretical guarantees, especially when compared to classical mesh-based methods. In this work, we focus on the inverse problem of identifying a spatially varying parameter in a constitutive model of three-dimensional elasticity, using measurements of the system's state. This setting is especially relevant for non-invasive diagnosis in cardiac biomechanics, where one must also carefully account for the type of boundary data available. To address this inverse problem, we adopt an all-at-once optimisation framework, simultaneously estimating the state and parameter through a least-squares loss that encodes both available data and the governing physics. For this formulation, we prove stability estimates ensuring that our approach yields a stable approximation of the underlying ground-truth parameter of the physical system independent of a specific discretisation. We then proceed with a neural network-based discretisation and compare it to traditional mesh-based approaches. Our theoretical findings are complemented by illustrative numerical examples.","short_abstract":"Physics-informed neural networks have emerged as a powerful tool in the scientific machine learning community, with applications to both forward and inverse problems. While they have shown considerable empirical success, significant challenges remain -- particularly regarding training stability and the lack of rigorous...","url_abs":"https://arxiv.org/abs/2512.09754","url_pdf":"https://arxiv.org/pdf/2512.09754v1","authors":"[\"Federica Caforio\",\"Martin Holler\",\"Matthias Höfler\"]","published":"2025-12-10T15:31:17Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.NA\"]","methods":"[]","has_code":false}