arXiv subjects

Bojana Rosić

Publications and source records attributed to Bojana Rosić.

9 recordsLinked to original sources

arXiv2026

A manifold-aware Neural ODE surrogate model for stochastic induction heating with anisotropic electrical conductivity

Induction welding plays a central role in enabling lightweight, integrated structures made from fibre-reinforced thermoplastic composites. From a modelling perspective, the induction welding process can be approximated by one-way coupled electromagnetic and heat-transfer equations. In practice, material parameters such as electrical conductivity vary significantly resulting from the deviations in the placement of fibres and hence the fibre-fibre contacts in the mesostructure are governed by consolidation quality of the material. Explicit representation of this variability on the macroscopic scale is essential to capture the closed current loops required for the induction heating process. To address this, a stochastic material model is introduced that respects the symmetric positive-definite (SPD) nature of the conductivity tensor and separates scaling and orientation uncertainties, forming the basis of a surrogate framework. The resulting stochastic conductivity model is first used to quantify the uncertainty in the induction heating process through extensive Monte Carlo simulations, providing detailed insight into the induced currents and the resulting temperature field. Subsequently, to enable efficient uncertainty propagation, SPD-aware surrogate models are trained with a subset of the simulation data, consisting of labelled material states and temperature fields. The surrogates are formulated as Constitutive Manifold Neural Networks (CMNNs) that explicitly respect the underlying SPD manifold structure and are integrated with a Neural Ordinary Differential Equation (NODE) framework to capture temporal dynamics. Several NODE integration schemes are evaluated and compared.

cs.CE↗

arXiv2026

Domain-Decomposition Neural Surrogates for Scalable Decentralized Ensemble Kalman Filter Based Parameter Identification in High-Dimensional Stochastic PDEs

Ensemble Kalman filters (EnKF) provide an efficient framework for parameter identification of physics based laws from spatially distributed measurements. Their forecast models require a large number of samples to accurately represent uncertainties, leading to high computational costs. A NN-based surrogate model is introduced to replace the sample-based forecast model. The proposed NN surrogate maps spatial coordinates and physics-based parameters to the forecasted observation. Such maps require a large number of parameters for high-dimensional spatial domains. To overcome this limitation, a augmented Lagrange multiplier domain decomposition method (DDM) is developed, where local NN models are optimized independently before global communication and coupling. This reduces the number of NN parameters while improving local approximation accuracy. Furthermore, a distributed and decentralized ensemble Kalman filter approach based on DDM-NN surrogate model is investigated, where the parameter identification problem is decomposed into local subproblems. Each local estimator updates the material parameters using locally available information, while communication between neighboring subdomains enables the reconstruction of a consistent global estimate to reduce the computational cost. The proposed method is evaluated on a three-dimensional material parameter identification problem and compared with an EnKF and a MCMC reference solution. The results show that the proposed DDM NN-based KF captures the posterior parameter distribution and approaches the solutions obtained with both EnKF and MCMC. While MCMC provides the most accurate representation of the posterior distribution, the proposed approach achieves comparable parameter estimates with reduced computational requirements for the forecast model.

cs.CE↗

arXiv2026

Generalized Kalman filter based temporal difference reinforcement learning

In this paper, we present a generalized temporal-difference (TD) reinforcement learning framework based on the theory of conditional expectations. The value and action-value (Q-value) functions are treated as uncertain quantities, and their estimation is formulated as a stochastic inference problem. Unlike classical Kalman-based temporal-difference learning, which relies on linear-Gaussian assumptions, the proposed formulation is derived directly from the conditional expectation framework and naturally extends to nonlinear models and non-Gaussian probability distributions. The proposed method recursively estimates not only the conditional expectation of the value function but also its second probabilistic moment, thereby quantifying the uncertainty associated with the learned value function throughout the learning process. To obtain a computationally tractable algorithm, the stochastic problem is discretized using either polynomial chaos expansions or ensemble-based approximations, providing efficient representations of the underlying random variables. The proposed framework is demonstrated on two optimal control problems: a linear mass--spring--damper system and a nonlinear heat conduction problem in a closed cavity. The numerical examples illustrate the capability of the proposed method to accurately estimate both the value function and its associated uncertainty, while extending classical Kalman-based temporal-difference learning to a broader class of stochastic systems.

cs.LG↗

arXiv2026

Domain decomposition of large neural network surrogate models

Neural networks (NNs) have gained significant attention across various engineering disciplines, particularly in design optimization, where they are used to build surrogate models for high-dimensional regression problems. Despite their power as global approximators, NNs often fail to accurately capture local nonlinearities without relying on a large number of training parameters. To address these limitations, in this paper we propose domain decomposition methods (DDM), which divide the input feature space into multiple local subdomains, each modeled by a simpler NN, trained in parallel. To recover the accuracy of a global approximation, interface constraints are introduced in the local loss functions to enforce continuity between subdomains. The interface constraints are enforced with two different approaches, by utilizing Lagrange multiplier or augmented Lagrange multiplier methods. Both approaches are validated using synthetic data from 2D and 3D linear compression problems, numerically solved using the finite element method. The study investigates computational time and accuracy across varying numbers of subdomains to identify optimal partitioning strategies. Compared to unconstrained approximations, both methods significantly improve continuity across subdomain interfaces. Also, the use of DDMs improves approximation accuracy in nonlinear regions when compared to standard global NN training. The augmented Lagrange method outperforms the standard Lagrange formulation by converging faster due to lower convergence requirements, albeit with a slightly lower accuracy. Its scalability makes it the preferred choice for large-scale problems, as the faster convergence outweighs the minor loss in accuracy. Overall, these results highlight the augmented Lagrange method as a promising DDM approach for training efficient and scalable NN surrogate models.

cs.CE↗

arXiv2025

Scale-invariant Monte Carlo and multilevel Monte Carlo estimation of mean and variance: An application to simulation of linear elastic bone tissue

We propose novel scale-invariant error estimators for the Monte Carlo and multilevel Monte Carlo estimation of mean and variance. For any linear transformation of the distribution of the quantity of interest, the computation cost across fidelity levels is optimized using a normalized error estimate, which is not only fully dimensionless but also remains robust to variation in characteristics of the distribution. We demonstrate the effectiveness of the algorithms through application to a mechanical simulation of linear elastic bone tissue, where material uncertainty incorporating both heterogeneity and random anisotropy is considered in the constitutive law.

math.NA↗

arXiv2025

Constitutive Manifold Neural Networks

Anisotropic material properties, such as the thermal conductivities of engineering composites, exhibit variability due to inherent material heterogeneity and manufacturing-related uncertainties. Mathematically, these properties are modeled as symmetric positive definite (SPD) tensors, which reside on a curved Riemannian manifold. Extending this description to a stochastic framework requires preserving both the SPD structure and the underlying spatial symmetries of the tensors. This is achieved through the spectral decomposition of tensors, which enables the parameterization of uncertainties into scale (strength) and rotation (orientation) components. To quantify the impact of strength and orientation uncertainties on the thermal behaviour of the composite, the stochastic material tensor must be propagated through a physics-based forward model. This process necessitates computationally efficient surrogate models, for which a feedforward neural network (FNN) is employed. However, conventional FNN architectures are not well-suited for SPD tensors, as directly using tensor components as input features fails to preserve their underlying geometric structure, often leading to suboptimal performance. To address this issue, we introduce the Constitutive Manifold Neural Network (CMNN), which incorporates input layers that map SPD tensors from the curved manifold to the local tangent space-a flat vector space-thus preserving the statistical and geometric information in the dataset. A case study involving steady-state heat conduction with stochastic anisotropic conductivity demonstrates that geometry-preserving neural network significantly enhances learning performance compared to conventional multilayer perceptrons (MLPs). These findings underscore the importance of manifold-aware methods when working with tensor-valued data in engineering applications.

cs.CE↗

arXiv2025

Stochastic Modelling of Elasticity Tensor Fields

We present a novel framework for the probabilistic modelling of random fourth order material tensor fields, with a focus on tensors that are physically symmetric and positive definite (SPD), of which the elasticity tensor is a prime example. Given the critical role that spatial symmetries and invariances play in determining material behaviour, it is essential to incorporate these aspects into the probabilistic description and modelling of material properties. In particular, we focus on spatial point symmetries or invariances under rotations, a classical subject in elasticity. Following this, we formulate a stochastic modelling framework using a Lie algebra representation via a memoryless transformation that respects the requirements of positive definiteness and invariance. With this, it is shown how to generate a random ensemble of elasticity tensors that allows an independent control of strength, eigenstrain, and orientation. The procedure also accommodates the requirement to prescribe specific spatial symmetries and invariances for each member of the whole ensemble, while ensuring that the mean or expected value of the ensemble conforms to a potentially 'higher' class of spatial invariance. Furthermore, it is important to highlight that the set of SPD tensors forms a differentiable manifold, which geometrically corresponds to an open cone within the ambient space of symmetric tensors. Thus, we explore the mathematical structure of the underlying sample space of such tensors, and introduce a new distance measure or metric, called the 'elasticity metric', between the tensors. Finally, we model and visualize a one-dimensional spatial field of orthotropic Kelvin matrices using interpolation based on the elasticity metric.

cs.CE↗

arXiv2024

Stochastic Modelling of Symmetric Positive Definite Material Tensors

Spatial symmetries and invariances play an important role in the behaviour of materials and should be respected in the description and modelling of material properties. The focus here is the class of physically symmetric and positive definite tensors, as they appear often in the description of materials, and one wants to be able to prescribe certain classes of spatial symmetries and invariances for each member of the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly 'higher' spatial invariance class. We formulate a modelling framework which not only respects these two requirements$-$positive definiteness and invariance$-$but also allows a fine control over orientation on one hand, and strength/size on the other. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear space of physically symmetric tensors, we consider it advantageous to widen the notion of mean to the so-called Fréchet mean on a metric space, which is based on distance measures or metrics between positive definite tensors other than the usual Euclidean one. It is shown how the random ensemble can be modelled and generated, independently in its scaling and orientational or directional aspects, with a Lie algebra representation via a memoryless transformation. The parameters which describe the elements in this Lie algebra are then to be considered as random fields on the domain of interest. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties$-$scaling, orientation, and prescribed material symmetry$-$on the desired quantities of interest.

math.NA↗

arXiv2019

Effect of Load Path on Parameter Identification for Plasticity Models using Bayesian Methods

To evaluate the cyclic behavior under different loading conditions using the kinematic and isotropic hardening theory of steel, a Chaboche viscoplastic material model is employed. The parameters of a constitutive model are usually identified by minimization of the distance between model response and experimental data. However, measurement errors and differences in the specimens lead to deviations in the determined parameters. In this article, the Choboche model is used and a stochastic simulation technique is applied to generate artificial data which exhibit the same stochastic behavior as experimental data. Then the model parameters are identified by applying an estimation using Bayes's theorem. The Gauss-Markov-Kalman filter using functional approximation is introduced and employed to estimate the model parameters in the Bayesian setting. Identified parameters are compared with the true parameters in the simulation, and the efficiency of the identification method is discussed. In the end, the effect of the load path on parameter identification is investigated.

cs.CE↗