arXiv subjects

Thomas J. R. Hughes

Publications and source records attributed to Thomas J. R. Hughes.

At least 19 recordsLinked to original sources

arXiv2026

Interpolating Neural Operator (INO): A Data-Free and Efficient Approach for Learning PDE Solution Operators

Neural operators have become a popular approach to approximate the solution operators of parametric partial differential equations (PDEs). However, existing neural operators either require a large amount of simulation data or a long physics-informed training on GPUs, and they cannot tell how accurate an individual prediction is. In this paper, we propose the Interpolating Neural Operator (INO), a data-free interpolating neural network that is trained directly on the weak form of the PDE. In INO, the Karhunen-Loève coordinates of the input field are treated as additional inputs together with the spatial coordinates, and each input is approximated by a C-HiDeNN sub-network whose trainable parameters are nodal values. Since the network is multilinear in its parameters, training reduces to a sequence of one-dimensional linear solves by greedy alternating least squares. As a result, INO trains on one CPU core and predicts a new solution in microseconds. For coercive problems, the total error of every prediction is bounded by a computable residual bound that requires no reference solution, and the same bound applies to the predictions of other methods that satisfy the boundary conditions exactly. Before each prediction, INO checks whether the leading coordinates of the input lie within the range on which it is trained, and inputs outside this range can be passed to a conventional solver or to an INO trained on a wider range. INO is compared with physics-informed FNO and DeepONet on different benchmarks. INO is the most accurate model on most of these problems, by 15$\times$ on two-dimensional Helmholtz at $65^2$ and 53$\times$ on the diffusion-reaction benchmark, and on the one- and two-dimensional problems its training on one CPU core takes 3-80$\times$ less time than the physics-informed baselines on one GPU.

cs.CE↗

arXiv2026

Geometry-Preserving Reduced-Order Modeling via Immersed Tensor Decomposition (ITD)

Body-fitted finite-element methods deliver high-order accuracy but hinge on a clean, watertight, conforming mesh, a requirement that breaks down for the geometrically imperfect CAD assemblies, image-based volumetric data, and voxel-native designs that pervade biomedical engineering and additive manufacturing, where mesh generation has become the dominant cost of the analysis cycle. Immersed methods on regular background Cartesian grids sidestep body-fitted meshing, but classical implementations integrate over irregular cut subdomains, destroying the tensor-product structure that enables separable, reduced-order methods such as tensor decomposition. In this paper we propose the \emph{Immersed Tensor Decomposition} (ITD) framework, which couples a mesh-free geometric representation via body-fitted function with the separable C-HiDeNN-TD reduced-order solver to enable large-scale simulation directly on regular background voxel meshes. The geometry is encoded in three steps: a signed-distance function represents the boundary, a body-fitted function $Φ$ approximates it with controllable error, and a low-rank Tucker decomposition provides model-order reduction; for a fixed grid spacing $h$, accuracy is improved by raising the approximation order of C-HiDeNN interpolation up to degree $p$ with a linear background mesh. The central contribution is an exact Dirichlet formulation that enforces the boundary condition strongly by multiplying the trial function with $Φ$, so that $u=g$ holds by construction without any variational penalty or interface quadrature. We establish an a priori error estimate for the formulation and assess it on canonical 2D/3D domains, demonstrating optimal convergence and robustness on non-Cartesian geometries discretized by regular voxel meshes.

math.NA↗

arXiv2026

Reconstruction of glymphatic transport fields from subject-specific imaging data, with particular emphasis on cerebrospinal fluid flow and tracer conservation

The reconstruction of physically valid transport fields from subject-specific imaging data is a fundamental challenge in image-based computational modeling due to measurement noise, modeling uncertainties and discretization errors. Without a methodology to construct models that faithfully reflect the underlying physics, mechanistic understanding of complex biological systems is inherently limited. In this work, we address this challenge in the glymphatic system, the brain's waste-clearance network, where cerebrospinal fluid (CSF) is transported through perivascular spaces into the brain parenchyma to facilitate metabolic waste removal. We introduce a computational framework for the high-fidelity reconstruction of subject-specific glymphatic transport fields from spatiotemporal imaging data. The formulation utilizes an advection-diffusion model with a velocity decomposition that imposes mass conservation, enabling the recovery of solenoidal (divergence-free) velocity fields through the solution of a constrained inverse problem. The system is discretized using immersed isogeometric analysis with quadratic B-spline basis functions, providing smooth, high-continuity solutions and inherent regularization of imaging noise. We demonstrate the framework's utility by using contrast-enhanced magnetic resonance imaging of tracer transport in a mouse brain, obtaining spatially varying estimates of CSF velocity, diffusivity, and clearance parameters. Forward simulations using the recovered fields show close agreement with experimental observations, validating the framework's ability to characterize complex transport dynamics while preserving physical integrity. This approach provides a generalizable methodology for the robust inference of physically consistent transport fields from imperfect imaging data, with broad applicability to the image-guided modeling of biological and engineering systems.

cs.CE↗

arXiv2026

Chemotaxis-inspired PDE models of airborne infectious disease transmission: epidemiologically-motivated mathematical and numerical analyses

Partial differential equation (PDE) models for infectious diseases, while less common than their ordinary differential equation (ODE) counterparts, have found successful applications for many years. Such models are typically of reaction-diffusion type, and model spatial propagation as a diffusive process. However, given the complex nature of human mobility, such models are limited in their ability to describe airborne infectious diseases in human populations. Recent work has advocated for the inclusion of an additional chemotaxis-type term as an alternative; spatial propagation of infection fronts is assumed additionally to flow from low-to-high concentrations of susceptible populations. The present work extends the study of such models by providing an epidemiologically interpretable analysis, directly connecting model behavior to information readily available to policymakers. In particular, we derive a spatially-aware basic reproduction number, which accounts for spatial heterogeneity in population density. Furthermore, we discuss several important aspects concerning the numerical solution of the model, including the introduction of a stabilization scheme. Finally, we perform a series of simulation studies in the Italian region of Lombardy (severely affected by the COVID-19 outbreak in 2020) and in the US state of Georgia, in which we demonstrate the model's potential to better capture important spatiotemporal dynamics observed in real-world data compared to pure reaction-diffusion models.

q-bio.PE↗

arXiv2025

MultiLevel Variational MultiScale (ML-VMS) framework for large-scale simulation

In this paper, we propose the MultiLevel Variational MultiScale (ML-VMS) method, a novel approach that seamlessly integrates a multilevel mesh strategy into the Variational Multiscale (VMS) framework. A key feature of the ML-VMS method is the use of the Convolutional Hierarchical Deep Neural Network (C-HiDeNN) as the approximation basis. The framework employs a coarse mesh throughout the domain, with localized fine meshes placed only in subdomains of high interest, such as those surrounding a source. Solutions at different resolutions are robustly coupled through the variational weak form and interface conditions. Compared to existing multilevel methods, ML-VMS (1) can couple an arbitrary number of mesh levels across different scales using variational multiscale framework; (2) allows approximating functions with arbitrary orders with linear finite element mesh due to the C-HiDeNN basis; (3) is supported by a rigorous theoretical error analysis; (4) features several tunable hyperparameters (e.g., order $p$, patch size $s$) with a systematic guide for their selection. We first show the theoretical error estimates of ML-VMS. Then through numerical examples, we demonstrate that ML-VMS with the C-HiDeNN takes less computational time than the FEM basis given comparable accuracy. Furthermore, we incorporate a space-time reduced-order model (ROM) based on C-HiDeNN-Tensor Decomposition (TD) into the ML-VMS framework. For a large-scale single-track laser powder bed fusion (LPBF) transient heat transfer problem that is equivalent to a full-order finite element model with $10^{10}$ spatial degrees of freedom (DoFs), our 3-level ML-VMS C-HiDeNN-TD achieves an approximately 5,000x speedup on a single CPU over a single-level linear FEM-TD ROM.

math.NA↗

arXiv2025

Phase-Field/Discontinuity Capturing operator for direct van der Waals simulation (DVS)

Discontinuity capturing (DC) operators are commonly employed to numerically solve problems involving sharp gradients in the solution. Despite their success, the application of DC operators to the direct van der Waals simulation (DVS) remains challenging. The DVS framework models non-equilibrium phase transitions by admitting interfacial regions in which the derivative of pressure with respect to density is negative. In these regions, we demonstrate that classical DC operators may violate the free energy dissipation law and produce unphysical wave structures. To address this limitation, we propose the phase-field/discontinuity capturing (PF/DC) operator. Numerical results show that PF/DC yields stable and accurate solutions in both bulk fluids and interfacial regions. Finally, we apply the proposed method to simulate cavitating flow over a three-dimensional bluff body, obtaining excellent agreement with experimental data and significant improvements over results produced using classical DC operators.

physics.comp-ph↗

arXiv2025

Large language model-empowered next-generation computer-aided engineering

Software development has entered a new era where large language models (LLMs) now serve as general-purpose reasoning engines, enabling natural language interaction and transformative applications across diverse domains. This paradigm is now extending into computer-aided engineering (CAE). Recent applications of LLMs in CAE have successfully automated routine tasks, including CAD model generation and FEM simulations. Nevertheless, these contributions, which primarily serve to reduce manual labor, are often insufficient for addressing the significant computational challenges posed by large-scale, high-dimensional systems. To this aim, we first introduce the concept of LLM-empowered CAE agent, where LLMs act as autonomous collaborators that plan, execute, and adapt CAE workflows. Then, we propose an LLM-empowered CAE agent for data-free model order reduction (MOR), a powerful yet underused approach for ultra-fast large-scale parametric analysis due to the intrusive nature and labor-intensive redevelopment of solvers. LLMs can alleviate this barrier by automating derivations, code restructuring, and implementation, making intrusive MOR both practical and broadly accessible. To demonstrate feasibility, we present an LLM-empowered CAE agent for solving ultra-large-scale space-parameter-time (S-P-T) physical problems using Tensor-decomposition-based A Priori Surrogates (TAPS). Our results show that natural language prompts describing parametric partial differential equations (PDEs) can be translated into efficient solver implementations, substantially reducing human effort while producing high-fidelity reduced-order models. Moreover, LLMs can synthesize novel MOR solvers for unseen cases such as nonlinear and high-dimensional parametric problems based on their internal knowledge base. This highlights the potential of LLMs to establish the foundation for next-generation CAE systems.

cs.CE↗

arXiv2025

A simple and efficient hybrid discretization approach to alleviate membrane locking in isogeometric thin shells

This work presents a new hybrid discretization approach to alleviate membrane locking in isogeometric finite element formulations for Kirchhoff-Love shells. The approach is simple, and requires no additional dofs and no static condensation. It does not increase the bandwidth of the tangent matrix and is effective for both linear and nonlinear problems. It combines isogeometric surface discretizations with classical Lagrange-based surface discretizations, and can thus be run with existing isogeometric finite element codes. Also, the stresses can be recovered straightforwardly. The effectiveness of the proposed approach in alleviating, if not eliminating, membrane locking is demonstrated through the rigorous study of the convergence behavior of several classical benchmark problems. Accuracy gains are particularly large in the membrane stresses. The approach is formulated here for quadratic NURBS, but an extension to other discretization types can be anticipated. The same applies to other constraints and associated locking phenomena.

cs.CE↗

arXiv2025

Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems

Deep learning has been extensively employed as a powerful function approximator for modeling physics-based problems described by partial differential equations (PDEs). Despite their popularity, standard deep learning models often demand prohibitively large computational resources and yield limited accuracy when scaling to large-scale, high-dimensional physical problems. Their black-box nature further hinders the application in industrial problems where interpretability and high precision are critical. To overcome these challenges, this paper introduces Interpolating Neural Network-Tensor Decomposition (INN-TD), a scalable and interpretable framework that has the merits of both machine learning and finite element methods for modeling large-scale physical systems. By integrating locally supported interpolation functions from finite element into the network architecture, INN-TD achieves a sparse learning structure with enhanced accuracy, faster training/solving speed, and reduced memory footprint. This makes it particularly effective for tackling large-scale high-dimensional parametric PDEs in training, solving, and inverse optimization tasks in physical problems where high precision is required.

cs.CE↗

arXiv2025

An Entropy Stable Formulation of Two-equation Turbulence Models with Particular Reference to the k-epsilon Model

Consistency and stability are two essential ingredients in the design of numerical algorithms for partial differential equations. Robust algorithms can be developed by incorporating nonlinear physical stability principles in their design, such as the entropy production inequality (i.e., the Clausius-Duhem inequality or second law of thermodynamics), rather than by simply adding artificial viscosity (a common approach). This idea is applied to the k-epsilon and two-equation turbulence models by introducing space-time averaging. Then, a set of entropy variables can be defined which leads to a symmetric system of advective-diffusive equations. Positivity and symmetry of the equations require certain constraints on the turbulence diffusivity coefficients and the turbulence source terms. With these, we are able to design entropy producing two-equation turbulence models and, in particular, the k-epsilon model.

cs.CE↗

arXiv2025

Tensor-decomposition-based A Priori Surrogate (TAPS) modeling for ultra large-scale simulations

A data-free, predictive scientific AI model, Tensor-decomposition-based A Priori Surrogate (TAPS), is proposed for tackling ultra large-scale engineering simulations with significant speedup, memory savings, and storage gain. TAPS can effectively obtain surrogate models for high-dimensional parametric problems with equivalent zetta-scale ($10^{21}$) degrees of freedom (DoFs). TAPS achieves this by directly obtaining reduced-order models through solving governing equations with multiple independent variables such as spatial coordinates, parameters, and time. The paper first introduces an AI-enhanced finite element-type interpolation function called convolution hierarchical deep-learning neural network (C-HiDeNN) with tensor decomposition (TD). Subsequently, the generalized space-parameter-time Galerkin weak form and the corresponding matrix form are derived. Through the choice of TAPS hyperparameters, an arbitrary convergence rate can be achieved. To show the capabilities of this framework, TAPS is then used to simulate a large-scale additive manufacturing process as an example and achieves around 1,370x speedup, 14.8x memory savings, and 955x storage gain compared to the finite difference method with $3.46$ billion spatial degrees of freedom (DoFs). As a result, the TAPS framework opens a new avenue for many challenging ultra large-scale engineering problems, such as additive manufacturing and integrated circuit design, among others.

cs.CE↗

arXiv2025

An optimal Petrov-Galerkin framework for operator networks

The optimal Petrov-Galerkin formulation to solve partial differential equations (PDEs) recovers the best approximation in a specified finite-dimensional (trial) space with respect to a suitable norm. However, the recovery of this optimal solution is contingent on being able to construct the optimal weighting functions associated with the trial basis. While explicit constructions are available for simple one- and two-dimensional problems, such constructions for a general multidimensional problem remain elusive. In the present work, we revisit the optimal Petrov-Galerkin formulation through the lens of deep learning. We propose an operator network framework called Petrov-Galerkin Variationally Mimetic Operator Network (PG-VarMiON), which emulates the optimal Petrov-Galerkin weak form of the underlying PDE. The PG-VarMiON is trained in a supervised manner using a labeled dataset comprising the PDE data and the corresponding PDE solution, with the training loss depending on the choice of the optimal norm. The special architecture of the PG-VarMiON allows it to implicitly learn the optimal weighting functions, thus endowing the proposed operator network with the ability to generalize well beyond the training set. We derive approximation error estimates for PG-VarMiON, highlighting the contributions of various error sources, particularly the error in learning the true weighting functions. Several numerical results are presented for the advection-diffusion equation to demonstrate the efficacy of the proposed method. By embedding the Petrov-Galerkin structure into the network architecture, PG-VarMiON exhibits greater robustness and improved generalization compared to other popular deep operator frameworks, particularly when the training data is limited.

math.NA↗

arXiv2023

Patient-specific computational forecasting of prostate cancer growth during active surveillance using an imaging-informed biomechanistic model

Active surveillance (AS) is a suitable management option for newly-diagnosed prostate cancer (PCa), which usually presents low to intermediate clinical risk. Patients enrolled in AS have their tumor closely monitored via longitudinal multiparametric magnetic resonance imaging (mpMRI), serum prostate-specific antigen tests, and biopsies. Hence, the patient is prescribed treatment when these tests identify progression to higher-risk PCa. However, current AS protocols rely on detecting tumor progression through direct observation according to standardized monitoring strategies. This approach limits the design of patient-specific AS plans and may lead to the late detection and treatment of tumor progression. Here, we propose to address these issues by leveraging personalized computational predictions of PCa growth. Our forecasts are obtained with a spatiotemporal biomechanistic model informed by patient-specific longitudinal mpMRI data. Our results show that our predictive technology can represent and forecast the global tumor burden for individual patients, achieving concordance correlation coefficients ranging from 0.93 to 0.99 across our cohort (n=7). Additionally, we identify a model-based biomarker of higher-risk PCa: the mean proliferation activity of the tumor (p=0.041). Using logistic regression, we construct a PCa risk classifier based on this biomarker that achieves an area under the receiver operating characteristic curve of 0.83. We further show that coupling our tumor forecasts with this PCa risk classifier enables the early identification of PCa progression to higher-risk disease by more than one year. Thus, we posit that our predictive technology constitutes a promising clinical decision-making tool to design personalized AS plans for PCa patients.

q-bio.TO↗

arXiv2023

Variationally Mimetic Operator Networks

In recent years operator networks have emerged as promising deep learning tools for approximating the solution to partial differential equations (PDEs). These networks map input functions that describe material properties, forcing functions and boundary data to the solution of a PDE. This work describes a new architecture for operator networks that mimics the form of the numerical solution obtained from an approximate variational or weak formulation of the problem. The application of these ideas to a generic elliptic PDE leads to a variationally mimetic operator network (VarMiON). Like the conventional Deep Operator Network (DeepONet) the VarMiON is also composed of a sub-network that constructs the basis functions for the output and another that constructs the coefficients for these basis functions. However, in contrast to the DeepONet, the architecture of these sub-networks in the VarMiON is precisely determined. An analysis of the error in the VarMiON solution reveals that it contains contributions from the error in the training data, the training error, the quadrature error in sampling input and output functions, and a "covering error" that measures the distance between the test input functions and the nearest functions in the training dataset. It also depends on the stability constants for the exact solution operator and its VarMiON approximation. The application of the VarMiON to a canonical elliptic PDE and a nonlinear PDE reveals that for approximately the same number of network parameters, on average the VarMiON incurs smaller errors than a standard DeepONet and a recently proposed multiple-input operator network (MIONet). Further, its performance is more robust to variations in input functions, the techniques used to sample the input and output functions, the techniques used to construct the basis functions, and the number of input functions.

math.NA↗

arXiv2021

Analysis-suitable unstructured T-splines: Multiple extraordinary points per face

Analysis-suitable T-splines (AST-splines) are a promising candidate to achieve a seamless integration between the design and the analysis of thin-walled structures in industrial settings. In this work, we generalize AST-splines to allow multiple extraordinary points within the same face. This generalization drastically increases the flexibility to build geometries using AST-splines; e.g., much coarser meshes can be generated to represent a certain geometry. The AST-spline spaces detailed in this work have $C^1$ inter-element continuity near extraordinary points and $C^2$ inter-element continuity elsewhere. We mathematically show that AST-splines with multiple extraordinary points per face are linearly independent and their polynomial basis functions form a non-negative partition of unity. We numerically show that AST-splines with multiple extraordinary points per face lead to optimal convergence rates for second- and fourth-order linear elliptic problems. To illustrate a possible isogeometric framework that is already available, we design the B-pillar and the side outer panel of a car using T-splines with the commercial software Autodesk Fusion360, import the control nets into our in-house code to build AST-splines, and import the Bézier extraction information into the commercial software LS-DYNA to solve eigenvalue problems. The results are compared with conventional finite elements. Good agreement is found, but conventional finite elements require significantly more degrees of freedom to reach a converged solution than AST-splines.

math.NA↗

arXiv2021

Quantitative in vivo imaging to enable tumor forecasting and treatment optimization

Current clinical decision-making in oncology relies on averages of large patient populations to both assess tumor status and treatment outcomes. However, cancers exhibit an inherent evolving heterogeneity that requires an individual approach based on rigorous and precise predictions of cancer growth and treatment response. To this end, we advocate the use of quantitative in vivo imaging data to calibrate mathematical models for the personalized forecasting of tumor development. In this chapter, we summarize the main data types available from both common and emerging in vivo medical imaging technologies, and how these data can be used to obtain patient-specific parameters for common mathematical models of cancer. We then outline computational methods designed to solve these models, thereby enabling their use for producing personalized tumor forecasts in silico, which, ultimately, can be used to not only predict response, but also optimize treatment. Finally, we discuss the main barriers to making the above paradigm a clinical reality.

q-bio.TO↗

arXiv2020

The Quad Layout Immersion: A Mathematically Equivalent Representation of a Surface Quadrilateral Layout

Quadrilateral layouts on surfaces are valuable in texture mapping, and essential in generation of quadrilateral meshes and in fitting splines. Previous work has characterized such layouts as a special metric on a surface or as a meromorphic quartic differential with finite trajectories. In this work, a surface quadrilateral layout is alternatively characterized as a special immersion of a cut representation of the surface into the Euclidean plane. We call this a quad layout immersion. This characterization, while posed in smooth topology, naturally generalizes to piecewise-linear representations. As such, it mathematically describes and generalizes integer grid maps, which are common in computer graphics settings. Finally, the utility of the representation is demonstrated by computationally extracting quadrilateral layouts on surfaces of interest.

cs.CG↗

arXiv2020

An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff-Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell's mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith's theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires $C^1$-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-$α$ method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton-Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.

cs.CE↗