Anthony Gruber

Adrienne M. Propp, Mauro Perego, Eric C. Cyr et al.

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.

Anthony Gruber, Irina Tezaur

A method for the nonintrusive and structure-preserving model reduction of canonical and noncanonical Hamiltonian systems is presented. Based on the idea of operator inference, this technique is provably convergent and reduces to a straightforward linear solve given snapshot data and gray-box knowledge of the system Hamiltonian. Examples involving several hyperbolic partial differential equations show that the proposed method yields reduced models which, in addition to being accurate and stable with respect to the addition of basis modes, preserve conserved quantities well outside the range of their training data.

Ian Moore, Christopher Wentland, Anthony Gruber et al.

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.

Arjun Vijaywargia, Eric C. Cyr, Anthony Gruber

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.

Xiaolong He, Yeonjong Shin, Anthony Gruber et al.

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.

Jonas A. Actor, Anthony Gruber, Eric C. Cyr

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.

Cheng Jing, Uvini Balasuriya Mudiyanselage, Woojin Cho et al.

Structure-preserving approaches to dynamics modeling have demonstrated great potential for modeling physical systems due to their strong inductive biases that enforce conservation laws and dissipative behavior. However, the resulting models are typically trained for fixed system configurations, requiring explicit knowledge of system parameters as well as costly retraining for each new set of parameters -- a major limitation in many-query or parameter-varying scenarios. Meta-learning offers a potential solution, but existing approaches like optimization-based meta-learning often suffer from training instability or limited generalization capability. Inspired by ideas from computer vision, we introduce a modulation-based meta-learning framework that directly conditions structure-preserving models on compact latent representations of potentially unknown system parameters, avoiding the need for gray-box system knowledge and explicit optimization during adaptation. Through the application of novel modulation strategies to parametric energy-conserving and dissipative systems, we enable scalable and generalizable learning across parametric families of dynamical systems. Experiments on standard benchmark problems demonstrate that our approach achieves accurate predictions in few-shot learning settings, without compromising on the essential physical constraints necessary for dynamical stability and effective generalization performance across parameter space.

Sanghyun Hong, Fan Wu, Anthony Gruber et al.

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.

Anthony Gruber, Kookjin Lee, Nathaniel Trask

Recent works have shown that physics-inspired architectures allow the training of deep graph neural networks (GNNs) without oversmoothing. The role of these physics is unclear, however, with successful examples of both reversible (e.g., Hamiltonian) and irreversible (e.g., diffusion) phenomena producing comparable results despite diametrically opposed mechanisms, and further complications arising due to empirical departures from mathematical theory. This work presents a series of novel GNN architectures based upon structure-preserving bracket-based dynamical systems, which are provably guaranteed to either conserve energy or generate positive dissipation with increasing depth. It is shown that the theoretically principled framework employed here allows for inherently explainable constructions, which contextualize departures from theory in current architectures and better elucidate the roles of reversibility and irreversibility in network performance.

Yuankai Teng, Zhu Wang, Lili Ju et al.

Due to the curse of dimensionality and the limitation on training data, approximating high-dimensional functions is a very challenging task even for powerful deep neural networks. Inspired by the Nonlinear Level set Learning (NLL) method that uses the reversible residual network (RevNet), in this paper we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples, whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and Active Subspace methods, especially when the target function possesses critical points in the interior of its input domain.

Anthony Gruber, Max Gunzburger, Lili Ju et al.

The popularity of deep convolutional autoencoders (CAEs) has engendered new and effective reduced-order models (ROMs) for the simulation of large-scale dynamical systems. Despite this, it is still unknown whether deep CAEs provide superior performance over established linear techniques or other network-based methods in all modeling scenarios. To elucidate this, the effect of autoencoder architecture on its associated ROM is studied through the comparison of deep CAEs against two alternatives: a simple fully connected autoencoder, and a novel graph convolutional autoencoder. Through benchmark experiments, it is shown that the superior autoencoder architecture for a given ROM application is highly dependent on the size of the latent space and the structure of the snapshot data, with the proposed architecture demonstrating benefits on data with irregular connectivity when the latent space is sufficiently large.

Anthony Gruber, Max Gunzburger, Lili Ju et al.

A dimension reduction method based on the "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.