Martin Holler

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.

Artificial intelligence offers great opportunities in critical care, particularly when a vast amount of continuously acquired physiological data is incorporated. High-quality, reliably labelled data are paramount for developing and training artificial intelligence methods. However, routinely recorded data in critical care are often noisy, and the sheer volume of high-resolution data is challenging to manage. Generalizable solutions for these problems are lacking, restricting progress. To address these barriers, we developed Vitabel , an open-source Python framework for post hoc loading, visualizing, aligning, and annotating medical time series. The framework provides sensible defaults and interactive components for efficient use in preconfigured workflows, while remaining flexible and extendable for custom analysis and annotation pipelines. It integrates seamlessly into Jupyter Notebooks , providing an interactive, customizable interface for visual interaction with the data. In this publication, we demonstrate its utility across three use cases. The code and exemplary data are provided as browser-based demos. Vitabel is freely available and published under the MIT license accompanying this publication.

This paper addresses the problem of learning reaction-diffusion (RD) systems from data while ensuring physical consistency and well-posedness of the learned models. Building on a regularization-based framework for structured model learning, we focus on learning parameterized reaction terms and investigate how to incorporate key physical properties, such as mass conservation and quasipositivity, directly into the learning process. Our main contributions are twofold: First, we propose techniques to systematically modify a given class of parameterized reaction terms such that the resulting terms inherently satisfy mass conservation and quasipositivity, ensuring that the learned RD systems preserve non-negativity and adhere to physical principles. These modifications also guarantee well-posedness of the resulting PDEs under additional regularity and growth conditions. Second, we extend existing theoretical results on regularization-based model learning to RD systems using these physically consistent reaction terms. Specifically, we prove that solutions to the learning problem converge to a unique, regularization-minimizing solution of a limit system even when conservation laws and quasipositivity are enforced. In addition, we provide approximation results for quasipositive functions, essential for constructing physically consistent parameterizations. These results advance the development of interpretable and reliable data-driven models for RD systems that align with fundamental physical laws.

Artificial intelligence offers a great opportunity in critical care, particularly when a vast amount of continuously acquired physiological data is incorporated. High-quality, reliably labelled data are paramount for developing and training artificial intelligence methods. However, routinely recorded data in critical care are often noisy, and the sheer volume of high-resolution data is challenging to manage. Generalizable solutions for these problems are lacking, restricting progress.To address these barriers, we developed \vitabel{}, an open-source \python{} package for loading, visualising, aligning, and annotating medical time series with minimal coding. The tool integrates seamlessly into \jupyter{-notebooks}, providing an interactive, customizable interface to interact with the data visually. In this publication, we demonstrate its utility across three use cases. The code and exemplary data are provided as browser-based demos. \vitabel{} is freely available and published under the MIT license accompanying this publication.

This paper addresses the problem of recovering tracer kinetic parameters from multi-region measurement data in quantitative PET imaging using the reversible two tissue compartment model. Its main result is an extension of our previous work on the irreversible two tissue compartment model. In analogy to our previous work, we show that also in the practically highly relevant reversible case, most tracer kinetic parameters can be uniquely identified from standard PET measurements without additional full blood sample analysis that is usually performed in practice, and under reasonable assumptions. In addition, unique identifiability of all parameters is shown, provided that additional measurements from the uncorrected total arterial blood tracer concentration are available. These can be obtained from standard PET measurements or from a simple blood sample analysis.

This paper addresses the problem of recovering tracer kinetic parameters from multi-region measurement data in quantitative PET imaging using the reversible two tissue compartment model. Its main result is an extension of our previous work on the irreversible two tissue compartment model. In analogy to our previous work, we show that also in the (practically highly relevant) reversible case, most tracer kinetic parameters can be uniquely identified from standard PET measurements (without additional full blood sample analysis that is usually performed in practice) and under reasonable assumptions. In addition, unique identifiability of all parameters is shown provided that additional measurements from the (uncorrected) total arterial blood tracer concentration (which can be obtained from standard PET measurements or from a simple blood sample analysis) are available.

This paper is concerned with sampling from probability distributions [Formula: see text] on [Formula: see text] admitting a density of the form [Formula: see text], where [Formula: see text], with [Formula: see text] being a linear operator and [Formula: see text] being nondifferentiable. Two different methods are proposed, both employing a subgradient step with respect to [Formula: see text] but, depending on the regularity of [Formula: see text], either an explicit or an implicit gradient step with respect to [Formula: see text]. For both methods, nonasymptotic convergence proofs are provided, with improved convergence results for more regular [Formula: see text]. Further, numerical experiments are conducted for simple 2D examples, illustrating the convergence rates, and for examples of Bayesian imaging, showing the practical feasibility of the proposed methods for high-dimensional data.