arXiv subjects

Anthony Gruber

Publications and source records attributed to Anthony Gruber.

At least 19 recordsLinked to original sources

arXiv2026

Vakonomic Fluids

We introduce a novel discretization of the incompressible Euler equations based on their interpretation as geodesic equations on the Lie group of volume-preserving diffeomorphisms. It is well known that encoding diffeomorphisms and their infinitesimal generators through a discretized Koopman representation places a nonholonomic constraint on discrete velocities, for which there is no consensus on a variational treatment. We show that taking the vakonomic perspective, as opposed to the usual perspective of Lagrange--d'Alembert, yields discrete fluid trajectories that remain geodesics on a (sub-)Riemannian manifold. In particular, the resulting vakonomic dynamics are Lie--Poisson and their solutions admit a discrete relabeling symmetry, leading to machine-precision satisfaction of Casimir invariants along with a discrete analogue of Kelvin's Circulation Theorem. Using an efficient momentum map representation based on low-rank Clebsch variables, we show that these vakonomic fluids behave stably and consistently even at low grid resolutions, leading to increased robustness and physical realism in the long term.

math-ph↗

arXiv2026

Flexible and Stable Dynamics Discovery with Onsager's Variational Principle

Variational principles specify the dynamics of a physical system via the extremization of associated functional data. Onsager's variational principle (OVP), which characterizes dissipation-dominated phenomena such as phase separation, admits an unconditionally energy-stable time discretization through the minimization of a Rayleighan functional combining free energy and dissipative effects. The present work considers the case where one or more parts of this functional are empirically approximated or otherwise uncertain. To address this, a novel variational discretization of OVP is introduced which recovers previous work as a special case, and a learning problem is formulated which identifies uncertain terms in the free energy and dissipation potential from observable data. It is shown that the resulting OVP-based models connect directly to previous work in terms of proximal methods, Sobolev and Wasserstein gradient flows, while remaining provably energy-stable under arbitrarily long rollouts. The approach is illustrated on examples including Allen-Cahn, Fokker-Planck, and Cahn-Hilliard system models, where the effects of bulk free-energy densities, nonlocal potentials, and nonstandard boundary conditions are effectively learned with model classes consisting of polynomials, shallow neural networks, and spectral convolution kernels.

math.DS↗

arXiv2026

Structure-Aware Tensorial Model Reduction

This work investigates a two-stage method for constructing projection-based reduced-order models (ROMs) of parameterized partial differential equations (PDEs). Based on established tensorial ROM methodology, the proposed approach reduces dimensionality offline by encoding solution snapshots using a multi-linear Tucker factorization, so that a reduced basis which varies nonlinearly with PDE parameters can be rapidly constructed online and used in a Galerkin ROM. Two novel extensions of this strategy, tailored to the cases of structured PDEs and sparse parameter sampling, are presented: the construction of reduced bases orthonormalized with respect to a general discrete inner product, and the interpolation of encoded states via radial basis functions. Basic representation and ROM error estimates are presented demonstrating the validity of these modifications, and the approach is challenged on examples where monolithic-basis ROMs are known to struggle, including a realistic instance of Maxwell's equations in 3D. Results suggest that the proposed nonlinear basis ROM can effectively mitigate linear restrictions on Kolmogorov $n$-width while improving upon previous tensorial ROM technology, particularly in the highly nonlinear and data-limited regimes characteristic of practical use cases.

math.NA↗

arXiv2026

NN-OpInf: an operator inference approach using structure-preserving composable neural networks

We propose neural network operator inference (NN-OpInf): a structure-preserving, composable, and minimally restrictive operator inference framework for the non-intrusive reduced-order modeling of dynamical systems. The approach learns latent dynamics from snapshot data, enforcing local operator structure such as skew-symmetry, (semi-)positive definiteness, and gradient preservation, while also reflecting complex dynamics by supporting additive compositions of heterogeneous operators. We present practical training strategies and analyze computational costs relative to linear and quadratic polynomial OpInf (P-OpInf). Numerical experiments across several nonlinear and parametric problems demonstrate improved accuracy, stability, and robustness over P-OpInf and prior NN-ROM formulations, particularly when the dynamics are not well represented by polynomial models. These results suggest that NN-OpInf can serve as an effective drop-in replacement for P-OpInf when the dynamics to be modeled contain non-polynomial nonlinearities, offering potential gains in accuracy and out-of-distribution performance at the expense of higher training computational costs and a more difficult, non-convex learning problem.

cs.LG↗

arXiv2025

Domain-Decomposed Graph Neural Network Surrogate Modeling for Ice Sheets

Accurate yet efficient surrogate models are essential for large-scale simulations of partial differential equations (PDEs), particularly for uncertainty quantification (UQ) tasks that demand hundreds or thousands of evaluations. We develop a physics-inspired graph neural network (GNN) surrogate that operates directly on unstructured meshes and leverages the flexibility of graph attention. To improve both training efficiency and generalization properties of the model, we introduce a domain decomposition (DD) strategy that partitions the mesh into subdomains, trains local GNN surrogates in parallel, and aggregates their predictions. We then employ transfer learning to fine-tune models across subdomains, accelerating training and improving accuracy in data-limited settings. Applied to ice sheet simulations, our approach accurately predicts full-field velocities on high-resolution meshes, substantially reduces training time relative to training a single global surrogate model, and provides a ripe foundation for UQ objectives. Our results demonstrate that graph-based DD, combined with transfer learning, provides a scalable and reliable pathway for training GNN surrogates on massive PDE-governed systems, with broad potential for application beyond ice sheet dynamics.

cs.LG↗

arXiv2025

Hybrid coupling with operator inference and the overlapping Schwarz alternating method

This paper presents a novel hybrid approach for coupling subdomain-local non-intrusive Operator Inference (OpInf) reduced order models (ROMs) with each other and with subdomain-local high-fidelity full order models (FOMs) with using the overlapping Schwarz alternating method (O-SAM). The proposed methodology addresses significant challenges in multiscale modeling and simulation, particularly the long runtime and complex mesh generation requirements associated with traditional high-fidelity simulations. By leveraging the flexibility of O-SAM, we enable the seamless integration of disparate models, meshes, and time integration schemes, enhancing computational efficiency while maintaining high accuracy. Our approach is demonstrated through a series of numerical experiments on complex three-dimensional (3D) solid dynamics problems, showcasing speedups of up to 106x compared to conventional FOM-FOM couplings. This work paves the way for more efficient simulation workflows in engineering applications, with potential extensions to a wide range of partial differential equations.

math.NA↗

arXiv2025

Domain Decomposition-Based Coupling of High-Fidelity Finite Element and Reduced Order Operator Inference Models Using the Schwarz Alternating Method

We propose a novel hybrid domain decomposition method that couples sub-domain-local high-fidelity finite element (FE) models with reduced order models (ROMs) using the Schwarz alternating method. By integrating the noninstrusive Operator Inference (OpInf) ROM, our approach accelerates the Schwarz process while allowing for geometry and mesh flexibility. We demonstrate the effectiveness of the new OpInf-FE method on a convection-dominated convection-diffusion-reaction problem, achieving stable and accurate predictive solutions while improving the ROM training process.

math.NA↗

arXiv2025

Transmission Conditions for the Non-Overlapping Schwarz Coupling of Full Order and Operator Inference Models

This work investigates transmission conditions for the domain decomposition-based coupling of subdomain-local models using the non-overlapping Schwarz alternating method (NO-SAM). Building on prior efforts involving overlapping SAM (O-SAM), we formulate and assess two NO-SAM variants, based on alternating Dirichlet-Neumann and Robin-Robin transmission conditions. For the subdomain-local models, we consider a mix of full order models (FOMs) and non-intrusive reduced order models (ROMs) constructed via an emerging model reduction technique known as operator inference (OpInf). Of particular novelty is the first application of NO-SAM to couple non-intrusive OpInf ROMs with each other, and with FOMs. Numerical studies on a one-dimensional linear elastic wave propagation benchmark problem demonstrate that transmission condition choice and parameter tuning significantly impact convergence rate, accuracy, and stability. Robin-Robin coupling often yields faster convergence than alternating Dirichlet-Neumann, though improper parameter selection can induce spurious oscillations at subdomain interfaces. For FOM-OpInf and OpInf-OpInf couplings, sufficient modal content in the ROM basis improves accuracy and mitigates instability, in some cases outperforming the coupled FOM-FOM reference solutions in both accuracy and efficiency. These findings highlight NO-SAM's potential for enabling flexible, non-intrusive, and efficient multi-model coupling across independently meshed subdomains, while emphasizing the need for careful interface condition design in higher-dimensional and predictive settings.

math.NA↗

arXiv2025

Deriving Transformer Architectures as Implicit Multinomial Regression

While attention has been empirically shown to improve model performance, it lacks a rigorous mathematical justification. This short paper establishes a novel connection between attention mechanisms and multinomial regression. Specifically, we show that in a fixed multinomial regression setting, optimizing over latent features yields solutions that align with the dynamics induced on features by attention blocks. In other words, the evolution of representations through a transformer can be interpreted as a trajectory that recovers the optimal features for classification.

cs.LG↗

arXiv2025

Meta-learning Structure-Preserving Dynamics

Structure-preserving approaches to dynamics discovery have demonstrated great potential for modeling physical systems due to their use of strong inductive biases, which enforce key features such as conservation laws and dissipative behavior. However, these models are typically trained on a per-configuration basis, requiring explicit knowledge of system parameters and costly retraining when these parameters vary. While meta-learning provides a potential remedy, optimization-based approaches can suffer from limited generalizability. Motivated by recent advances in modulation-based learning aimed at mitigating these drawbacks, we systematically investigate the use of modulation techniques in learning conservative dynamical systems. We study a range of existing modulation strategies alongside newly proposed variants, integrating them into a Hamiltonian learning framework without requiring an explicit system parameterization. Through extensive experiments on benchmark problems, we demonstrate that modulation-based meta-learning enables accurate few-shot adaptation, achieving robust generalization across parameter space without compromising the conservation of key invariants responsible for the dynamics.

cs.LG↗

arXiv2025

Thermodynamically Consistent Latent Dynamics Identification for Parametric Systems

We propose an efficient thermodynamics-informed latent space dynamics identification (tLaSDI) framework for the reduced-order modeling of parametric nonlinear dynamical systems. This framework integrates autoencoders for dimensionality reduction with newly developed parametric GENERIC formalism-informed neural networks (pGFINNs), which enable efficient learning of parametric latent dynamics while preserving key thermodynamic principles such as free energy conservation and entropy generation across the parameter space. To further enhance model performance, a physics-informed active learning strategy is incorporated, leveraging a greedy, residual-based error indicator to adaptively sample informative training data, outperforming uniform sampling at equivalent computational cost. Numerical experiments on the Burgers' equation and the 1D/1V Vlasov-Poisson equation demonstrate that the proposed method achieves up to 3,528x speed-up with 1-3% relative errors, and significant reduction in training (50-90%) and inference (57-61%) cost. Moreover, the learned latent space dynamics reveal the underlying thermodynamic behavior of the system, offering valuable insights into the physical-space dynamics.

cs.LG↗

arXiv2025

Tensor parametric Hamiltonian operator inference

This work presents a tensorial approach to constructing data-driven reduced-order models corresponding to semi-discrete partial differential equations with canonical Hamiltonian structure. By expressing parameter-varying operators with affine dependence as contractions of a generalized parameter vector against a constant tensor, this method leverages the operator inference framework to capture parametric dependence in the learned reduced-order model via the solution to a convex, least-squares optimization problem. This leads to a concise and straightforward implementation which compactifies previous parametric operator inference approaches and directly extends to learning parametric operators with symmetry constraints, a key feature required for constructing structure-preserving surrogates of Hamiltonian systems. The proposed approach is demonstrated on both a (non-Hamiltonian) heat equation with variable diffusion coefficient as well as a Hamiltonian wave equation with variable wave speed.

math.NA↗

arXiv2025

Modeling Neural Networks with Privacy Using Neural Stochastic Differential Equations

In this work, we study the feasibility of using neural ordinary differential equations (NODEs) to model systems with intrinsic privacy properties. Unlike conventional feedforward neural networks, which have unlimited expressivity and can represent arbitrary mappings between inputs and outputs, NODEs constrain their learning to the solution of a system of differential equations. We first examine whether this constraint reduces memorization and, consequently, the membership inference risks associated with NODEs. We conduct a comprehensive evaluation of NODEs under membership inference attacks and show that they exhibit twice the resistance compared to conventional models such as ResNets. By analyzing the variance in membership risks across different NODE models, we find that their limited expressivity leads to reduced overfitting to the training data. We then demonstrate, both theoretically and empirically, that membership inference risks can be further mitigated by utilizing a stochastic variant of NODEs: neural stochastic differential equations (NSDEs). We show that NSDEs are differentially-private (DP) learners that provide the same provable privacy guarantees as DPSGD, the de-facto mechanism for training private models. NSDEs are also effective in mitigating membership inference attacks, achieving risk levels comparable to private models trained with DP-SGD while offering an improved privacyutility trade-off. Moreover, we propose a drop-in-replacement strategy that efficiently integrates NSDEs into conventional feedforward architectures to enhance their privacy.

cs.CR↗

arXiv2024

MaD-Scientist: AI-based Scientist solving Convection-Diffusion-Reaction Equations Using Massive PINN-Based Prior Data

Large language models (LLMs), like ChatGPT, have shown that even trained with noisy prior data, they can generalize effectively to new tasks through in-context learning (ICL) and pre-training techniques. Motivated by this, we explore whether a similar approach can be applied to scientific foundation models (SFMs). Our methodology is structured as follows: (i) we collect low-cost physics-informed neural network (PINN)-based approximated prior data in the form of solutions to partial differential equations (PDEs) constructed through an arbitrary linear combination of mathematical dictionaries; (ii) we utilize Transformer architectures with self and cross-attention mechanisms to predict PDE solutions without knowledge of the governing equations in a zero-shot setting; (iii) we provide experimental evidence on the one-dimensional convection-diffusion-reaction equation, which demonstrate that pre-training remains robust even with approximated prior data, with only marginal impacts on test accuracy. Notably, this finding opens the path to pre-training SFMs with realistic, low-cost data instead of (or in conjunction with) numerical high-cost data. These results support the conjecture that SFMs can improve in a manner similar to LLMs, where fully cleaning the vast set of sentences crawled from the Internet is nearly impossible.

cs.LG↗

arXiv2024

Gaussian Variational Schemes on Bounded and Unbounded Domains

A machine-learnable variational scheme using Gaussian radial basis functions (GRBFs) is presented and used to approximate linear problems on bounded and unbounded domains. In contrast to standard mesh-free methods, which use GRBFs to discretize strong-form differential equations, this work exploits the relationship between integrals of GRBFs, their derivatives, and polynomial moments to produce exact quadrature formulae which enable weak-form expressions. Combined with trainable GRBF means and covariances, this leads to a flexible, generalized Galerkin variational framework which is applied in the infinite-domain setting where the scheme is conforming, as well as the bounded-domain setting where it is not. Error rates for the proposed GRBF scheme are derived in each case, and examples are presented demonstrating utility of this approach as a surrogate modeling technique.

math.NA↗

arXiv2024

Domain Decomposition-based coupling of Operator Inference reduced order models via the Schwarz alternating method

This paper presents and evaluates an approach for coupling together subdomain-local reduced order models (ROMs) constructed via non-intrusive operator inference (OpInf) with each other and with subdomain-local full order models (FOMs), following a domain decomposition of the spatial geometry on which a given partial differential equation (PDE) is posed. Joining subdomain-local models is accomplished using the overlapping Schwarz alternating method, a minimally-intrusive multiscale coupling technique that works by transforming a monolithic problem into a sequence of subdomain-local problems, which communicate through transmission boundary conditions imposed on the subdomain interfaces. After formulating the overlapping Schwarz alternating method for OpInf ROMs, termed OpInf-Schwarz, we evaluate the method's accuracy and efficiency on several test cases involving the heat equation in two spatial dimensions. We demonstrate that the method is capable of coupling together arbitrary combinations of OpInf ROMs and FOMs, and that speed-ups over a monolithic FOM are possible when performing OpInf ROM coupling.

math.NA↗

arXiv2024

Efficiently Parameterized Neural Metriplectic Systems

Metriplectic systems are learned from data in a way that scales quadratically in both the size of the state and the rank of the metriplectic data. Besides being provably energy conserving and entropy stable, the proposed approach comes with approximation results demonstrating its ability to accurately learn metriplectic dynamics from data as well as an error estimate indicating its potential for generalization to unseen timescales when approximation error is low. Examples are provided which illustrate performance in the presence of both full state information as well as when entropic variables are unknown, confirming that the proposed approach exhibits superior accuracy and scalability without compromising on model expressivity.

cs.LG↗

arXiv2024

Variationally consistent Hamiltonian model reduction

Though ubiquitous as first-principles models for conservative phenomena, Hamiltonian systems present numerous challenges for model reduction even in relatively simple, linear cases. Here, we present a method for the projection-based model reduction of canonical Hamiltonian systems that is variationally consistent for any choice of linear reduced basis: Hamiltonian models project to Hamiltonian models. Applicable in both intrusive and nonintrusive settings, the proposed method is energy-conserving and symplectic, with error provably decomposable into a data projection term and a term measuring deviation from canonical form. Examples from linear elasticity with realistic material parameters are used to demonstrate the advantages of a variationally consistent approach, highlighting the steady convergence exhibited by consistent models where previous methods reliant on inconsistent techniques or specially designed bases exhibit unacceptably large errors.

math.NA↗