arXiv subjects

Bahador Bahmani

Publications and source records attributed to Bahador Bahmani.

16 recordsLinked to original sources

arXiv2026

Data-Driven Cohesive Zone Modeling within the Generalized Standard Materials Framework

Cohesive zone models are widely used to describe fracture and interfacial failure, yet most formulations prescribe problem-specific analytical traction-separation laws together with phenomenological rules for unloading and reloading, leading to specialized models for different cohesive behaviors. This work develops a unified learnable cohesive formulation within the generalized standard materials framework, in which the response is generated from learned constitutive functions while the underlying thermodynamic structure remains fixed. The surface free energy is decomposed into active and contact contributions, and irreversible damage evolution is governed by a learned mode-dependent damage resistance. The active energy is represented by an input-convex neural network, while the inverse damage resistance is represented by a monotone neural network. Convexity, monotonicity, normalization, and damage irreversibility are incorporated directly into the constitutive representation. Direct parameterization of the inverse resistance yields an explicit damage update and avoids local nonlinear inversion during constitutive evaluation. Material-point studies show that the formulation can represent qualitatively distinct cohesive responses, including plateaus, extended softening tails, irregular softening, nonlinear unloading, distinct Mode I and Mode II behaviors, and several classical mixed-mode cohesive laws. The framework therefore replaces law-specific model construction with a single thermodynamically structured representation capable of learning cohesive responses of broad functional complexity from data.

physics.comp-ph↗

arXiv2026

Constitutive Priors for Inverse Design

With recent advances in material synthesis and additive manufacturing, material systems can be designed to achieve prescribed mechanical responses. An important class of such problems is the inverse design of elastic networks that attain a target configuration under loading, with applications in robotics, aerospace, and shape-morphing structures. This work presents a framework that formulates inverse design directly within the learned family of constitutive behaviors. Given a collection of stress-strain responses, we construct a constitutive prior, defined as a low-dimensional latent representation of admissible material laws learned directly from these responses. Spatially varying latent variables are then optimized subject to the governing equilibrium equations so that the deformed network matches a target configuration. The constitutive prior is represented using an energy-based, partially input-convex neural network that enforces the constitutive constraints by construction. To improve robustness in the resulting nonconvex optimization problem, the framework combines homotopy-continuation-based optimization with correspondence-free point cloud matching, allowing the target and optimized geometries to have different discretizations. The proposed approach is demonstrated on several inverse design problems for nonlinear elastic networks, and quantitative comparisons with alternative optimization strategies show improved robustness and optimization performance.

physics.comp-ph↗

arXiv2026

Neural Operator Representation of Granular Micromechanics-based Failure Envelope

Micromechanics-based granular models are widely used to predict the failure behavior of porous and particulate materials, including concrete, soils, foams, and biological tissues. Although these models offer considerable flexibility through microstructural parametrization and statistical representation, their mapping to macroscopic responses, particularly failure envelopes, is implicit and requires costly nonlinear, non-smooth simulations, where each failure point is obtained by following a loading trajectory. This limitation is further amplified in inverse settings, where one seeks microstructure configurations that reproduce a target failure response. In this work, we propose a differentiable neural operator that learns the mapping from microstructure configurations to failure envelopes, enabling efficient forward prediction and inverse identification without repeated micromechanical simulations. To ensure mechanical admissibility, we incorporate a physics-informed training strategy that enforces convexity of the predicted envelopes, consistent with Drucker's postulate, thereby eliminating potential non-physical artifacts. We also compare finite difference and automatic differentiation for evaluating the proposed regularization, and find that finite difference provides a favorable practical trade-off in the present DeepONet-based setting. The operator is trained on failure envelopes represented as irregular point clouds, allowing learning from data sampled at heterogeneous resolutions. To further reduce computational cost, we introduce an active learning strategy that adaptively queries the micromechanical model in regions of high epistemic uncertainty. This leads to efficient exploration of the parameter space with fewer high-fidelity simulations. The versatility and performance of the method are demonstrated and benchmarked through several numerical examples.

physics.comp-ph↗

arXiv2026

Conformal Quantile Regression for Neural Probabilistic Constitutive Modeling

Biological soft tissues exhibit substantial inter-subject variability, making the automation of constitutive material modeling essential for patient-specific analysis and design. Such materials are not only highly nonlinear but also display intrinsic stochasticity arising from their complex and heterogeneous microstructure. Despite recent advances in data-driven constitutive modeling, most existing approaches remain deterministic and fail to quantify predictive uncertainty, thereby limiting their reliability in downstream mechanical analyses. In this work, we propose a probabilistic, data-driven constitutive modeling framework for anisotropic soft materials that explicitly accounts for uncertainty through conformalized quantile regression applied to tensor-valued fields. The proposed framework is built upon a strain-invariant, polyconvex formulation that ensures thermodynamic consistency and promotes robust predictive performance, including in extrapolative regimes. A key advantage of the proposed approach is its simplicity: it can be applied in a plug-and-play manner to endow existing deterministic models with probabilistic predictions, while remaining distribution-free and requiring no assumptions on the underlying data distribution. Moreover, the method is straightforward to train, scalable to models with a large number of parameters, and avoids Monte Carlo sampling at inference, making it computationally efficient and well suited for uncertainty propagation in large-scale mechanical simulations. The proposed method is validated using several benchmark datasets synthesized and collected from the literature.

physics.comp-ph↗

arXiv2026

A Multimodal Conditional Mixture Model with Distribution-Level Physics Priors

Many scientific and engineering systems exhibit intrinsically multimodal behavior arising from latent regime switching and non-unique physical mechanisms. In such settings, learning the full conditional distribution of admissible outcomes in a physically consistent and interpretable manner remains a challenge. While recent advances in machine learning have enabled powerful multimodal generative modeling, their integration with physics-constrained scientific modeling remains nontrivial, particularly when physical structure must be preserved or data are limited. This work develops a physics-informed multimodal conditional modeling framework based on mixture density representations. Mixture density networks (MDNs) provide an explicit and interpretable parameterization of multimodal conditional distributions. Physical knowledge is embedded through component-specific regularization terms that penalize violations of governing equations or physical laws. This formulation naturally accommodates non-uniqueness and stochasticity while remaining computationally efficient and amenable to conditioning on contextual inputs. The proposed framework is evaluated across a range of scientific problems in which multimodality arises from intrinsic physical mechanisms rather than observational noise, including bifurcation phenomena in nonlinear dynamical systems, stochastic partial differential equations, and atomistic-scale shock dynamics. In addition, the proposed method is compared with a conditional flow matching (CFM) model, a representative state-of-the-art generative modeling approach, demonstrating that MDNs can achieve competitive performance while offering a simpler and more interpretable formulation.

cs.LG↗

arXiv2026

A Multi-Fidelity Bayesian Neural Operator for Mechanics of Spinodal Metamaterial

Cellular metamaterials offer a vast design space for tailoring nonlinear mechanical responses, yet exploring this space with conventional modeling approaches is often infeasible or not scalable. To fully exploit their nonlinear behavior for inverse design, it is essential to learn the full stress-strain response rather than relying on bulk quantities, motivating the use of neural operators for function-to-function mapping. However, data-driven modeling of nonlinear response for metamaterials is severely constrained by the limited availability of costly experimental data. Here, we propose a Bayesian multi-fidelity deep operator network that aggregates abundant low-fidelity finite element simulations with sparse high-fidelity experimental data from in-situ nanomechanical experiments on spinodal metamaterials, enabling heterogeneous information aggregation. A hybrid Bayesian active learning strategy is introduced to select informative samples by jointly maximizing epistemic uncertainty and geometric diversity of the microstructure, substantially reducing the cost of 3D nonlinear simulations. This approach adaptively trains the low-fidelity operator, which is then augmented by a high-fidelity Bayesian residual learner. We demonstrate that only 22 strategically selected samples from a design pool of 3000 are sufficient to achieve an 84.1 percent reduction in MSE compared to the high-fidelity baseline. The framework significantly outperforms single-fidelity baselines, providing superior predictions for full nonlinear stress-strain responses as well as stiffness, strength, and energy absorption. This work provides a robust, data-efficient pathway for the inverse design and constitutive modeling of cellular metamaterials.

physics.app-ph↗

arXiv2025

A Physics-informed Multi-resolution Neural Operator

The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to obtain in some real-world engineering applications. These datasets may be unevenly discretized from one realization to another, with the grid resolution varying across samples. In this study, we introduce a physics-informed operator learning approach by extending the Resolution Independent Neural Operator (RINO) framework to a fully data-free setup, addressing both challenges simultaneously. Here, the arbitrarily (but sufficiently finely) discretized input functions are projected onto a latent embedding space (i.e., a vector space of finite dimensions), using pre-trained basis functions. The operator associated with the underlying partial differential equations (PDEs) is then approximated by a simple multi-layer perceptron (MLP), which takes as input a latent code along with spatiotemporal coordinates to produce the solution in the physical space. The PDEs are enforced via a finite difference solver in the physical space. The validation and performance of the proposed method are benchmarked on several numerical examples with multi-resolution data, where input functions are sampled at varying resolutions, including both coarse and fine discretizations.

cs.LG↗

arXiv2024

A Resolution Independent Neural Operator

The Deep Operator Network (DeepONet) is a powerful neural operator architecture that uses two neural networks to map between infinite-dimensional function spaces. This architecture allows for the evaluation of the solution field at any location within the domain but requires input functions to be discretized at identical locations, limiting practical applications. We introduce a general framework for operator learning from input-output data with arbitrary sensor locations and counts. This begins by introducing a resolution-independent DeepONet (RI-DeepONet), which handles input functions discretized arbitrarily but sufficiently finely. To achieve this, we propose two dictionary learning algorithms that adaptively learn continuous basis functions, parameterized as implicit neural representations (INRs), from correlated signals on arbitrary point clouds. These basis functions project input function data onto a finite-dimensional embedding space, making it compatible with DeepONet without architectural changes. We specifically use sinusoidal representation networks (SIRENs) as trainable INR basis functions. Similarly, the dictionary learning algorithms identify basis functions for output data, defining a new neural operator architecture: the Resolution Independent Neural Operator (RINO). In RINO, the operator learning task reduces to mapping coefficients of input basis functions to output basis functions. We demonstrate RINO's robustness and applicability in handling arbitrarily sampled input and output functions during both training and inference through several numerical examples.

cs.LG↗

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

Physics-constrained symbolic model discovery for polyconvex incompressible hyperelastic materials

We present a machine learning framework capable of consistently inferring mathematical expressions of hyperelastic energy functionals for incompressible materials from sparse experimental data and physical laws. To achieve this goal, we propose a polyconvex neural additive model (PNAM) that enables us to express the hyperelastic model in a learnable feature space while enforcing polyconvexity. An upshot of this feature space obtained via the PNAM is that (1) it is spanned by a set of univariate basis that can be re-parametrized with a more complex mathematical form, and (2) the resultant elasticity model is guaranteed to fulfill the polyconvexity, which ensures that the acoustic tensor remains elliptic for any deformation. To further improve the interpretability, we use genetic programming to convert each univariate basis into a compact mathematical expression. The resultant multi-variable mathematical models obtained from this proposed framework are not only more interpretable but are also proven to fulfill physical laws. By controlling the compactness of the learned symbolic form, the machine learning-generated mathematical model also requires fewer arithmetic operations than its deep neural network counterparts during deployment. This latter attribute is crucial for scaling large-scale simulations where the constitutive responses of every integration point must be updated within each incremental time step. We compare our proposed model discovery framework against other state-of-the-art alternatives to assess the robustness and efficiency of the training algorithms and examine the trade-off between interpretability, accuracy, and precision of the learned symbolic hyperelastic models obtained from different approaches. Our numerical results suggest that our approach extrapolates well outside the training data regime due to the precise incorporation of physics-based knowledge.

cs.CE↗

arXiv2024

Discovering interpretable elastoplasticity models via the neural polynomial method enabled symbolic regressions

Conventional neural network elastoplasticity models are often perceived as lacking interpretability. This paper introduces a two-step machine learning approach that returns mathematical models interpretable by human experts. In particular, we introduce a surrogate model where yield surfaces are expressed in terms of a set of single-variable feature mappings obtained from supervised learning. A post-processing step is then used to re-interpret the set of single-variable neural network mapping functions into mathematical form through symbolic regression. This divide-and-conquer approach provides several important advantages. First, it enables us to overcome the scaling issue of symbolic regression algorithms. From a practical perspective, it enhances the portability of learned models for partial differential equation solvers written in different programming languages. Finally, it enables us to have a concrete understanding of the attributes of the materials, such as convexity and symmetries of models, through automated derivations and reasoning. Numerical examples have been provided, along with an open-source code to enable third-party validation.

cs.CE↗

arXiv2021

Manifold embedding data-driven mechanics

This article introduces a new data-driven approach that leverages a manifold embedding generated by the invertible neural network to improve the robustness, efficiency, and accuracy of the constitutive-law-free simulations with limited data. We achieve this by training a deep neural network to globally map data from the constitutive manifold onto a lower-dimensional Euclidean vector space. As such, we establish the relation between the norm of the mapped Euclidean vector space and the metric of the manifold and lead to a more physically consistent notion of distance for the material data. This treatment in return allows us to bypass the expensive combinatorial optimization, which may significantly speed up the model-free simulations when data are abundant and of high dimensions. Meanwhile, the learning of embedding also improves the robustness of the algorithm when the data is sparse or distributed unevenly in the parametric space. Numerical experiments are provided to demonstrate and measure the performance of the manifold embedding technique under different circumstances. Results obtained from the proposed method and those obtained via the classical energy norms are compared.

cs.LG↗

arXiv2021

Equivariant geometric learning for digital rock physics: estimating formation factor and effective permeability tensors from Morse graph

We present a SE(3)-equivariant graph neural network (GNN) approach that directly predicting the formation factor and effective permeability from micro-CT images. FFT solvers are established to compute both the formation factor and effective permeability, while the topology and geometry of the pore space are represented by a persistence-based Morse graph. Together, they constitute the database for training, validating, and testing the neural networks. While the graph and Euclidean convolutional approaches both employ neural networks to generate low-dimensional latent space to represent the features of the micro-structures for forward predictions, the SE(3) equivariant neural network is found to generate more accurate predictions, especially when the training data is limited. Numerical experiments have also shown that the new SE(3) approach leads to predictions that fulfill the material frame indifference whereas the predictions from classical convolutional neural networks (CNN) may suffer from spurious dependence on the coordinate system of the training data. Comparisons among predictions inferred from training the CNN and those from graph convolutional neural networks (GNN) with and without the equivariant constraint indicate that the equivariant graph neural network seems to perform better than the CNN and GNN without enforcing equivariant constraints.

physics.geo-ph↗

arXiv2021

Training multi-objective/multi-task collocation physics-informed neural network with student/teachers transfer learnings

This paper presents a PINN training framework that employs (1) pre-training steps that accelerates and improve the robustness of the training of physics-informed neural network with auxiliary data stored in point clouds, (2) a net-to-net knowledge transfer algorithm that improves the weight initialization of the neural network and (3) a multi-objective optimization algorithm that may improve the performance of a physical-informed neural network with competing constraints. We consider the training and transfer and multi-task learning of physics-informed neural network (PINN) as multi-objective problems where the physics constraints such as the governing equation, boundary conditions, thermodynamic inequality, symmetry, and invariant properties, as well as point cloud used for pre-training can sometimes lead to conflicts and necessitating the seek of the Pareto optimal solution. In these situations, weighted norms commonly used to handle multiple constraints may lead to poor performance, while other multi-objective algorithms may scale poorly with increasing dimensionality. To overcome this technical barrier, we adopt the concept of vectorized objective function and modify a gradient descent approach to handle the issue of conflicting gradients. Numerical experiments are compared the benchmark boundary value problems solved via PINN. The performance of the proposed paradigm is compared against the classical equal-weighted norm approach. Our numerical experiments indicate that the brittleness and lack of robustness demonstrated in some PINN implementations can be overcome with the proposed strategy.

cs.LG↗

arXiv2021

Data-driven discovery of interpretable causal relations for deep learning material laws with uncertainty propagation

This paper presents a computational framework that generates ensemble predictive mechanics models with uncertainty quantification (UQ). We first develop a causal discovery algorithm to infer causal relations among time-history data measured during each representative volume element (RVE) simulation through a directed acyclic graph (DAG). With multiple plausible sets of causal relationships estimated from multiple RVE simulations, the predictions are propagated in the derived causal graph while using a deep neural network equipped with dropout layers as a Bayesian approximation for uncertainty quantification. We select two representative numerical examples (traction-separation laws for frictional interfaces, elastoplasticity models for granular assembles) to examine the accuracy and robustness of the proposed causal discovery method for the common material law predictions in civil engineering applications.

cs.LG↗

arXiv2020

An accelerated hybrid data-driven/model-based approach for poroelasticity problems with multi-fidelity multi-physics data

We present a hybrid model/model-free data-driven approach to solve poroelasticity problems. Extending the data-driven modeling framework originated from Kirchdoerfer and Ortiz (2016), we introduce one model-free and two hybrid model-based/data-driven formulations capable of simulating the coupled diffusion-deformation of fluid-infiltrating porous media with different amounts of available data. To improve the efficiency of the model-free data search, we introduce a distance-minimized algorithm accelerated by a k-dimensional tree search. To handle the different fidelities of the solid elasticity and fluid hydraulic constitutive responses, we introduce a hybridized model in which either the solid and the fluid solver can switch from a model-based to a model-free approach depending on the availability and the properties of the data. Numerical experiments are designed to verify the implementation and compare the performance of the proposed model to other alternatives.

cs.LG↗