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Moritz Flaschel

Publications and source records attributed to Moritz Flaschel.

At least 19 recordsLinked to original sources
arXiv2026

Benchmarking data-driven material models on the classic Treloar dataset

Machine learning is rapidly reshaping constitutive modeling, offers new ways to learn material behavior directly from experimental data, and challenges long-established modeling paradigms. But with a growing number of machine-learning-based approaches available, how do they compare in practice? In this paper, we use the classic experimental data of Treloar to benchmark popular frameworks for hyperelasticity: (Generalized-Invariant) Constitutive Artificial Neural Networks, Physics-Augmented Neural Networks, (Adaptive) Material Fingerprinting, and Efficient Unsupervised Constitutive Law Identification & Discovery. We compare their fitting performance, computational cost, hyperparameter sensitivity, and ease of implementation. Furthermore, we discuss the trade-offs between predictive accuracy and model complexity. The latter is assessed by quantifying both the number of material parameters in the discovered models and the computational time required to evaluate the constitutive model and its derivatives. The results show that all methods can reproduce the benchmark data remarkably well. Rather than identifying a single winner, we highlight the strengths and limitations of each approach and provide practical guidance for their use. The source code for all six methods, including the training and comparison scripts, as well as all results and data used in this study, is publicly available via https://doi.org/10.5281/zenodo.21915635.

cs.AI↗
arXiv2026

Neural operators solve inverse problems for constitutive model discovery

Characterizing the mechanical response of materials traditionally requires solving optimization problems in which model parameters are calibrated or trained to minimize the discrepancy between model predictions and experimental data. This process can be computationally expensive and time-consuming. To overcome this limitation, we propose two neural operator architectures that directly map experimentally measured data to the constitutive functions governing the mechanical response of the material: Physics-Augmented Neural Operators (PANO) and Constitutive Artificial Neural Operators (CANO). The proposed neural operators approximate the mapping between the infinite-dimensional input space of full-field displacement measurements and net reaction forces, and the infinite-dimensional output space of hyperelastic strain-energy density functions. The displacement fields are encoded through Laplacian eigenfunctions to obtain discretization-independent and noise-robust predictions. Our framework constrains the output space to physically admissible material models that satisfy fundamental physical requirements by design. The neural operators are trained on simulated data tuples of displacement fields and reaction forces for a range of material models. Once trained, the neural operators enable near-instantaneous material characterization and require only a single forward pass to infer the strain-energy density function from a given experimental dataset. We test the predictive power of the neural operators for unseen data, noisy data, data with missing information, data from different spatial discretizations, and data from geometries of different sizes.

cs.CE↗
arXiv2026

Adaptive Material Fingerprinting for the fast discovery of polyconvex feature combinations in isotropic and anisotropic hyperelasticity

We recently proposed a method called Material Fingerprinting for the rapid discovery of mechanical material models that avoids solving continuous optimization problems. Material Fingerprinting assumes that each material exhibits a unique response when subjected to a standardized experimental setup, which is interpreted as the material's mechanical fingerprint. If a database of fingerprints is generated in an offline phase, a model for an unseen experimental measurement can be discovered in real time by comparing the experimentally measured fingerprint to the fingerprints in the database. In our original contributions, the database comprised a fixed number of material models, each with a fixed number of parameters. To increase the fitting flexibility of Material Fingerprinting, we propose an adaptive model database coupled with an iterative pattern recognition algorithm that refines the material model in each step. This strategy enables Material Fingerprinting to discover arbitrary linear combinations of material models from the database, rather than being restricted to selecting a single model from a predefined set. In comparison to previous works on Material Fingerprinting, this enables the discovery of more complex models, such as multi-term Ogden models or the anisotropic Holzapfel-Gasser-Ogden model. To design the adaptive database, we leverage sums of strain energy density feature functions that depend on isotropic and anisotropic invariants. All modeling features satisfy fundamental physical constraints, and polyconvexity can be optionally enforced via a simple user-controlled switch. We test the method on experimental data stemming from mechanical tests of isotropic rubber materials and anisotropic animal skin tissue.

cs.CE↗
arXiv2026

Infinite-Dimensional Closed-Loop Inverse Kinematics for Soft Robots via Neural Operators

For fully actuated rigid robots, kinematic inversion is a purely geometric problem, efficiently solved by closed-loop inverse kinematics (CLIK) schemes that compute joint configurations to position the robot body in space. For underactuated soft robots, however, not all configurations are attainable through control action, making kinematic inversion extremely challenging. Extensions of CLIK address this by introducing end-to-end mappings from actuation to task space for the controller to operate on, but typically assume finite dimensions of the underlying virtual configuration space. In this work, we formulate CLIK in the infinite-dimensional domain to reason about the entire soft robot shape while solving tasks. We do this by composing an actuation-to-shape map with a shape-to-task map, deriving the differential end-to-end kinematics via an infinite-dimensional chain rule, and thereby obtaining a Jacobian-based CLIK algorithm. Since this actuation-to-shape mapping is rarely available in closed form, we propose to learn it using differentiable neural operator networks. We first present an analytical study on a constant-curvature segment, and then apply the neural version of the algorithm to a three-fiber soft robotic arm whose underlying model relies on morphoelasticity and active filament theory.

cs.RO↗
arXiv2026

Unsupervised Material Fingerprinting: Ultra-fast hyperelastic model discovery from full-field experimental measurements

Material Fingerprinting is a lookup table-based strategy to discover material models from experimental measurements, which completely avoids the need to solve an optimization problem. In an offline phase, a comprehensive database of simulated material responses, so-called material fingerprints, is generated for a predefined experimental setup. This database can then be used repeatedly in the online phase to discover material models corresponding to experimentally measured observations. To this end, the experimentally measured fingerprint is compared with all fingerprints in the database to identify the closest match. The primary advantage of this strategy is that it does not require solving a continuous optimization problem. This avoids the associated computational costs as well as issues of ill-posedness caused by local minima in non-convex optimization landscapes. Material Fingerprinting has been successfully demonstrated for supervised datasets consisting of stress-strain pairs, as well as for unsupervised datasets involving full-field displacements and net reaction forces. However, to date, there is no experimental validation for the latter approach which is the objective of this work.

cs.CE↗
arXiv2025

Discovery of Hyperelastic Constitutive Laws from Experimental Data with EUCLID

We assess the performance of EUCLID, Efficient Unsupervised Constitutive Law Identification and Discovery, a recently proposed framework for automated discovery of constitutive laws, on experimental data. Mechanical tests are performed on natural rubber specimens spanning simple to complex geometries, from which we collect both global, force elongation, and local, full-field displacement, measurements. Using these data, we obtain constitutive laws via two routes, the conventional identification of unknown parameters in a priori selected material models, and EUCLID, which automates model selection and parameter identification within a unified model-discovery pipeline. We compare the two methodologies using global versus local data, analyze predictive accuracy, and examine generalization to unseen geometries. Moreover, we quantify the experimental noise, investigate the coverage of the material state space achieved by each approach and discuss the relative performance of different datasets and different a priori chosen models versus EUCLID.

physics.comp-ph↗
arXiv2025

Shape-Space Graphs: Fast and Collision-Free Path Planning for Soft Robots

Soft robots, inspired by elephant trunks or octopus arms, offer extraordinary flexibility to bend, twist, and elongate in ways that rigid robots cannot. However, their motion planning remains a challenge, especially in cluttered environments with obstacles, due to their highly nonlinear and infinite-dimensional kinematics. Here, we present a graph-based path planning tool for an elephant-trunk-inspired soft robot designed with three artificial muscle fibers that allow for continuous deformation through contraction. Using a biomechanical model that integrates morphoelastic and active filament theories, we precompute a shape library and construct a k-nearest neighbor graph in \emph{shape space}, ensuring that each node corresponds to a valid robot shape. For the graph, we use signed distance functions to prune nodes and edges colliding with obstacles, and define multi-objective edge costs based on geometric distance and actuation effort, enabling energy-aware planning with collision avoidance. We demonstrate that our algorithm reliably avoids obstacles and generates feasible paths within milliseconds from precomputed graphs using Dijkstra's algorithm. We show that including energy costs can drastically reduce the actuation effort compared to geometry-only planning, at the expense of longer tip trajectories. Our results highlight the potential of shape-space graph search for fast and reliable path planning in the field of soft robotics, paving the way for real-time applications in surgical, industrial, and assistive settings.

cs.RO↗
arXiv2025

Material Fingerprinting: A shortcut to material model discovery without solving optimization problems

We propose Material Fingerprinting, a new method for the rapid discovery of mechanical material models from direct or indirect data that avoids solving potentially non-convex optimization problems. The core assumption of Material Fingerprinting is that each material exhibits a unique response when subjected to a standardized experimental setup. We can interpret this response as the material's fingerprint, essentially a unique identifier that encodes all pertinent information about the material's mechanical characteristics. Consequently, once we have established a database containing fingerprints and their corresponding mechanical models during an offline phase, we can rapidly characterize an unseen material in an online phase. This is accomplished by measuring its fingerprint and employing a pattern recognition algorithm to identify the best matching fingerprint in the database. In our study, we explore this concept in the context of hyperelastic materials, demonstrating the applicability of Material Fingerprinting across different experimental setups. Initially, we examine Material Fingerprinting through experiments involving homogeneous deformation fields, which provide direct strain-stress data pairs. We then extend this concept to experiments involving complexly shaped specimens with heterogeneous deformation fields, which provide indirect displacement and reaction force measurements. We show that, in both cases, Material Fingerprinting is an efficient tool for model discovery, bypassing the challenges of potentially non-convex optimization. We believe that Material Fingerprinting provides a powerful and generalizable framework for rapid material model identification across a wide range of experimental designs and material behaviors, paving the way for numerous future developments.

cs.CE↗
arXiv2025

Non-smooth optimization meets automated material model discovery

Automated material model discovery disrupts the tedious and time-consuming cycle of iteratively calibrating and modifying manually designed models. Non-smooth L1-norm regularization is the backbone of automated model discovery; however, the current literature on automated material model discovery offers limited insights into the robust and efficient minimization of non-smooth objective functions. In this work, we examine the minimization of functions of the form f(w) + a ||w||_1, where w are the material model parameters, f is a metric that quantifies the mismatch between the material model and the observed data, and a is a regularization parameter that determines the sparsity of the solution. We investigate both the straightforward case where f is quadratic and the more complex scenario where it is non-quadratic or even non-convex. Importantly, we do not only focus on methods that solve the sparse regression problem for a given value of the regularization parameter a, but propose methods to efficiently compute the entire regularization path, facilitating the selection of a suitable a. Specifically, we present four algorithms and discuss their roles for automated material model discovery in mechanics: First, we recapitulate a well-known coordinate descent algorithm that solves the minimization problem assuming that f is quadratic for a given value of a, also known as the LASSO. Second, we discuss the algorithm LARS, which automatically determines the critical values of a, at which material parameters in w are set to zero. Third, we propose to use the proximal gradient method ISTA for automated material model discovery if f is not quadratic, and fourth, we suggest a pathwise extension of ISTA for computing the regularization path. We demonstrate the applicability of all algorithms for the discovery of hyperelastic material models from uniaxial tension and simple shear data.

cs.CE↗
arXiv2025

A topology optimisation framework to design test specimens for one-shot identification or discovery of material models

The increasing availability of full-field displacement data from imaging techniques in experimental mechanics is determining a gradual shift in the paradigm of material model calibration and discovery, from using several simple-geometry tests towards a few, or even one single test with complicated geometry. The feasibility of such a "one-shot" calibration or discovery heavily relies upon the richness of the measured displacement data, i.e., their ability to probe the space of the state variables and the stress space (whereby the stresses depend on the constitutive law being sought) to an extent sufficient for an accurate and robust calibration or discovery process. The richness of the displacement data is in turn directly governed by the specimen geometry. In this paper, we propose a density-based topology optimisation framework to optimally design the geometry of the target specimen for calibration of an anisotropic elastic material model. To this end, we perform automatic, high-resolution specimen design by maximising the robustness of the solution of the inverse problem, i.e., the identified material parameters, given noisy displacement measurements from digital image correlation. We discuss the choice of the cost function and the design of the topology optimisation framework, and we analyse a range of optimised topologies generated for the identification of isotropic and anisotropic elastic responses.

cs.CE↗
arXiv2024

A review on data-driven constitutive laws for solids

This review article highlights state-of-the-art data-driven techniques to discover, encode, surrogate, or emulate constitutive laws that describe the path-independent and path-dependent response of solids. Our objective is to provide an organized taxonomy to a large spectrum of methodologies developed in the past decades and to discuss the benefits and drawbacks of the various techniques for interpreting and forecasting mechanics behavior across different scales. Distinguishing between machine-learning-based and model-free methods, we further categorize approaches based on their interpretability and on their learning process/type of required data, while discussing the key problems of generalization and trustworthiness. We attempt to provide a road map of how these can be reconciled in a data-availability-aware context. We also touch upon relevant aspects such as data sampling techniques, design of experiments, verification, and validation.

cs.CE↗
arXiv2024

Reduced and All-at-Once Approaches for Model Calibration and Discovery in Computational Solid Mechanics

In the framework of solid mechanics, the task of deriving material parameters from experimental data has recently re-emerged with the progress in full-field measurement capabilities and the renewed advances of machine learning. In this context, new methods such as the virtual fields method and physics-informed neural networks have been developed as alternatives to the already established least-squares and finite element-based approaches. Moreover, model discovery problems are starting to emerge and can also be addressed in a parameter estimation framework. These developments call for a new unified perspective, which is able to cover both traditional parameter estimation methods and novel approaches in which the state variables or the model structure itself are inferred as well. Adopting concepts discussed in the inverse problems community, we distinguish between all-at-once and reduced approaches. With this general framework, we are able to structure a large portion of the literature on parameter estimation in computational mechanics - and we can identify combinations that have not yet been addressed, two of which are proposed in this paper. We also discuss statistical approaches to quantify the uncertainty related to the estimated parameters, and we propose a novel two-step procedure for identification of complex material models based on both frequentist and Bayesian principles. Finally, we illustrate and compare several of the aforementioned methods with mechanical benchmarks based on synthetic and real data.

cs.CE↗
arXiv2024

Data-driven methods for quantitative imaging

In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative Magnetic Resonance Imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps.

math.OC↗
arXiv2023

Automated discovery of interpretable hyperelastic material models for human brain tissue with EUCLID

We propose an automated computational algorithm for simultaneous model selection and parameter identification for the hyperelastic mechanical characterization of human brain tissue. Following the motive of the recently proposed computational framework EUCLID (Efficient Unsupervised Constitutive Law Identitication and Discovery) and in contrast to conventional parameter calibration methods, we construct an extensive set of candidate hyperelastic models, i.e., a model library including popular models known from the literature, and develop a computational strategy for automatically selecting a model from the library that conforms to the available experimental data while being represented as an interpretable symbolic mathematical expression. This computational strategy comprises sparse regression, i.e., a regression problem that is regularized by a sparsity promoting penalty term that filters out irrelevant models from the model library, and a clustering method for grouping together highly correlated and thus redundant features in the model library. The model selection procedure is driven by labelled data pairs stemming from mechanical tests under different deformation modes, i.e., uniaxial compression/tension and simple torsion, and can thus be interpreted as a supervised counterpart to the originally proposed EUCLID that is informed by full-field displacement data and global reaction forces. The proposed method is verified on synthetical data with artificial noise and validated on experimental data acquired through mechanical tests of human brain specimens, proving that the method is capable of discovering hyperelastic models that exhibit both high fitting accuracy to the data as well as concise and thus interpretable mathematical representations.

q-bio.QM↗
arXiv2023

Automated identification of linear viscoelastic constitutive laws with EUCLID

We extend EUCLID, a computational strategy for automated material model discovery and identification, to linear viscoelasticity. For this case, we perform a priori model selection by adopting a generalized Maxwell model expressed by a Prony series, and deploy EUCLID for identification. The methodology is based on four ingredients: i. full-field displacement and net force data; ii. a very wide material model library - in our case, a very large number of terms in the Prony series; iii. the linear momentum balance constraint; iv. the sparsity constraint. The devised strategy comprises two stages. Stage 1 relies on sparse regression; it enforces momentum balance on the data and exploits sparsity-promoting regularization to drastically reduce the number of terms in the Prony series and identify the material parameters. Stage 2 relies on k-means clustering; starting from the reduced set of terms from stage 1, it further reduces their number by grouping together Maxwell elements with very close relaxation times and summing the corresponding moduli. Automated procedures are proposed for the choice of the regularization parameter in stage 1 and of the number of clusters in stage 2. The overall strategy is demonstrated on artificial numerical data, both without and with the addition of noise, and shown to efficiently and accurately identify a linear viscoelastic model with five relaxation times across four orders of magnitude, out of a library with several hundreds of terms spanning relaxation times across seven orders of magnitude.

cond-mat.mtrl-sci↗
arXiv2023

Single-test evaluation of directional elastic properties of anisotropic structured materials

When the elastic properties of structured materials become direction-dependent, the number of their descriptors increases. For example, in two-dimensions, the anisotropic behavior of materials is described by up to 6 independent elastic stiffness parameters, as opposed to only 2 needed for isotropic materials. Such high number of parameters expands the design space of structured materials and leads to unusual phenomena, such as materials that can shear under uniaxial compression. However, an increased number of properties descriptors and the coupling between shear and normal deformations render the experimental evaluation of material properties more challenging. In this paper, we propose a methodology based on the virtual fields method to identify six separate stiffness tensor parameters of two-dimensional anisotropic structured materials using just one tension test, thus eliminating the need for multiple experiments, as it is typical in traditional methods. The approach requires no stress data and uses full-field displacement data and global force data. We show the accuracy of our method using synthetic data generated from finite element simulations as well as experimental data from additively manufactured specimens

cond-mat.mtrl-sci↗
arXiv2022

Automated discovery of generalized standard material models with EUCLID

We extend the scope of our approach for unsupervised automated discovery of material laws (EUCLID) to the case of a material belonging to an unknown class of behavior. To this end, we leverage the theory of generalized standard materials, which encompasses a plethora of important constitutive classes. We show that, based only on full-field kinematic measurements and net reaction forces, EUCLID is able to automatically discover the two scalar thermodynamic potentials, namely, the Helmholtz free energy and the dissipation potential, which completely define the behavior of generalized standard materials. The a priori enforced constraint of convexity on these potentials guarantees by construction stability and thermodynamic consistency of the discovered model; balance of linear momentum acts as a fundamental constraint to replace the availability of stress-strain labeled pairs; sparsity promoting regularization enables the automatic selection of a small subset from a possibly large number of candidate model features and thus leads to a parsimonious, i.e., simple and interpretable, model. Importantly, since model features go hand in hand with the correspondingly active internal variables, sparse regression automatically induces a parsimonious selection of the few internal variables needed for an accurate but simple description of the material behavior. A fully automatic procedure leads to the selection of the hyperparameter controlling the weight of the sparsity promoting regularization term, in order to strike a user-defined balance between model accuracy and simplicity. By testing the method on synthetic data including artificial noise, we demonstrate that EUCLID is able to automatically discover the true hidden material model from a large catalog of constitutive classes, including elasticity, viscoelasticity, elastoplasticity, viscoplasticity, isotropic and kinematic hardening.

cond-mat.mtrl-sci↗
arXiv2022

Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties

Within the scope of our recent approach for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID), we propose an unsupervised Bayesian learning framework for discovery of parsimonious and interpretable constitutive laws with quantifiable uncertainties. As in deterministic EUCLID, we do not resort to stress data, but only to realistically measurable full-field displacement and global reaction force data; as opposed to calibration of an a priori assumed model, we start with a constitutive model ansatz based on a large catalog of candidate functional features; we leverage domain knowledge by including features based on existing, both physics-based and phenomenological, constitutive models. In the new Bayesian-EUCLID approach, we use a hierarchical Bayesian model with sparsity-promoting priors and Monte Carlo sampling to efficiently solve the parsimonious model selection task and discover physically consistent constitutive equations in the form of multivariate multi-modal probabilistic distributions. We demonstrate the ability to accurately and efficiently recover isotropic and anisotropic hyperelastic models like the Neo-Hookean, Isihara, Gent-Thomas, Arruda-Boyce, Ogden, and Holzapfel models in both elastostatics and elastodynamics. The discovered constitutive models are reliable under both epistemic uncertainties - i.e. uncertainties on the true features of the constitutive catalog - and aleatoric uncertainties - which arise from the noise in the displacement field data, and are automatically estimated by the hierarchical Bayesian model.

cs.CE↗