Panos Stinis

Oded Ovadia, Adar Kahana, Panos Stinis et al.

We combine vision transformers with operator learning to solve diverse inverse problems described by partial differential equations (PDEs). Our approach, named ViTO, combines a U-Net based architecture with a vision transformer. We apply ViTO to solve inverse PDE problems of increasing complexity, namely for the wave equation, the Navier-Stokes equations and the Darcy equation. We focus on the more challenging case of super-resolution, where the input dataset for the inverse problem is at a significantly coarser resolution than the output. The results we obtain are comparable or exceed the leading operator network benchmarks in terms of accuracy. Furthermore, ViTO`s architecture has a small number of trainable parameters (less than 10% of the leading competitor), resulting in a performance speed-up of over 5x when averaged over the various test cases.

Zhi-Feng Wei, Saad Qadeer, Panos Stinis

Reduced-order modeling of high-dimensional dynamical systems is often hindered by the non-Markovian closure term that represents the effect of unresolved variables on the resolved dynamics. Inspired by the Mori--Zwanzig formalism, in which the closure takes the form of a memory functional of the resolved trajectory, we recast closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework: a Mamba-based sequence model, trained to predict the closure from the resolved trajectory, is coupled with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. A key feature of the framework is its exploitation of the dual representation of state-space models -- the model is trained in a sequence-to-sequence fashion via the convolutional form, and deployed for step-by-step autoregressive rollout via the recurrent form, yielding both efficient long-trajectory training and constant per-step inference cost. On the viscous Burgers' equation and the chaotic two-scale Lorenz '96 system, the MAC model substantially outperforms the Markovian reduced-order model, the GRU-based sequence model, and the Wilks method in predictive accuracy and long-time rollout stability.

QiZhi He, Mauro Perego, Amanda A. Howard et al.

One of the most challenging and consequential problems in climate modeling is to provide probabilistic projections of sea level rise. A large part of the uncertainty of sea level projections is due to uncertainty in ice sheet dynamics. At the moment, accurate quantification of the uncertainty is hindered by the cost of ice sheet computational models. In this work, we develop a hybrid approach to approximate existing ice sheet computational models at a fraction of their cost. Our approach consists of replacing the finite element model for the momentum equations for the ice velocity, the most expensive part of an ice sheet model, with a Deep Operator Network, while retaining a classic finite element discretization for the evolution of the ice thickness. We show that the resulting hybrid model is very accurate and it is an order of magnitude faster than the traditional finite element model. Further, a distinctive feature of the proposed model compared to other neural network approaches, is that it can handle high-dimensional parameter spaces (parameter fields) such as the basal friction at the bed of the glacier, and can therefore be used for generating samples for uncertainty quantification. We study the impact of hyper-parameters, number of unknowns and correlation length of the parameter distribution on the training and accuracy of the Deep Operator Network on a synthetic ice sheet model. We then target the evolution of the Humboldt glacier in Greenland and show that our hybrid model can provide accurate statistics of the glacier mass loss and can be effectively used to accelerate the quantification of uncertainty.

Qian Zhang, Adar Kahana, George Em Karniadakis et al.

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for long time integration of time-dependent Ordinary and Partial Differential Equations (ODEs, PDEs). The core element of the method is a SNN, trained to use spike-encoded information about the solution at previous timesteps to predict spike-encoded information at the next timestep. After the network has been trained, it operates as an explicit numerical scheme that can be used to compute the solution at future timesteps, given a spike-encoded initial condition. A decoder is used to transform the evolved spiking-encoded solution back to function values. We present results from numerical experiments of using the proposed method for ODEs and PDEs of varying complexity.

QiZhi He, Yucheng Fu, Panos Stinis et al.

Numerical modeling and simulation have become indispensable tools for advancing a comprehensive understanding of the underlying mechanisms and cost-effective process optimization and control of flow batteries. In this study, we propose an enhanced version of the physics-constrained deep neural network (PCDNN) approach [1] to provide high-accuracy voltage predictions in the vanadium redox flow batteries (VRFBs). The purpose of the PCDNN approach is to enforce the physics-based zero-dimensional (0D) VRFB model in a neural network to assure model generalization for various battery operation conditions. Limited by the simplifications of the 0D model, the PCDNN cannot capture sharp voltage changes in the extreme SOC regions. To improve the accuracy of voltage prediction at extreme ranges, we introduce a second (enhanced) DNN to mitigate the prediction errors carried from the 0D model itself and call the resulting approach enhanced PCDNN (ePCDNN). By comparing the model prediction with experimental data, we demonstrate that the ePCDNN approach can accurately capture the voltage response throughout the charge--discharge cycle, including the tail region of the voltage discharge curve. Compared to the standard PCDNN, the prediction accuracy of the ePCDNN is significantly improved. The loss function for training the ePCDNN is designed to be flexible by adjusting the weights of the physics-constrained DNN and the enhanced DNN. This allows the ePCDNN framework to be transferable to battery systems with variable physical model fidelity.

Amanda A. Howard, Mauro Perego, George E. Karniadakis et al.

Operator learning for complex nonlinear systems is increasingly common in modeling multi-physics and multi-scale systems. However, training such high-dimensional operators requires a large amount of expensive, high-fidelity data, either from experiments or simulations. In this work, we present a composite Deep Operator Network (DeepONet) for learning using two datasets with different levels of fidelity to accurately learn complex operators when sufficient high-fidelity data is not available. Additionally, we demonstrate that the presence of low-fidelity data can improve the predictions of physics-informed learning with DeepONets. We demonstrate the new multi-fidelity training in diverse examples, including modeling of the ice-sheet dynamics of the Humboldt glacier, Greenland, using two different fidelity models and also using the same physical model at two different resolutions.

Panos Stinis, Constantinos Daskalakis, Paul J. Atzberger

We introduce adversarial learning methods for data-driven generative modeling of the dynamics of $n^{th}$-order stochastic systems. Our approach builds on Generative Adversarial Networks (GANs) with generative model classes based on stable $m$-step stochastic numerical integrators. We introduce different formulations and training methods for learning models of stochastic dynamics based on observation of trajectory samples. We develop approaches using discriminators based on Maximum Mean Discrepancy (MMD), training protocols using conditional and marginal distributions, and methods for learning dynamic responses over different time-scales. We show how our approaches can be used for modeling physical systems to learn force-laws, damping coefficients, and noise-related parameters. The adversarial learning approaches provide methods for obtaining stable generative models for dynamic tasks including long-time prediction and developing simulations for stochastic systems.

Amanda A. Howard, Nicholas Zolman, Bruno Jacob et al.

Kolmogorov-Arnold networks (KANs) have arisen as a potential way to enhance the interpretability of machine learning. However, solutions learned by KANs are not necessarily interpretable, in the sense of being sparse or parsimonious. Sparse identification of nonlinear dynamics (SINDy) is a complementary approach that allows for learning sparse equations for dynamical systems from data; however, learned equations are limited by the library. In this work, we present SINDy-KANs, which simultaneously train a KAN and a SINDy-like representation to increase interpretability of KAN representations with SINDy applied at the level of each activation function, while maintaining the function compositions possible through deep KANs. We apply our method to a number of symbolic regression tasks, including dynamical systems, to show accurate equation discovery across a range of systems.

Huan Lei, Jing Li, Peiyuan Gao et al.

The challenge of quantifying uncertainty propagation in real-world systems is rooted in the high-dimensionality of the stochastic input and the frequent lack of explicit knowledge of its probability distribution. Traditional approaches show limitations for such problems. To address these difficulties, we have developed a general framework of constructing surrogate models on spaces of stochastic input with arbitrary probability measure irrespective of the mutual dependencies between individual components and the analytical form. The present Data-driven Sparsity-enhancing Rotation for Arbitrary Randomness (DSRAR) framework includes a data-driven construction of multivariate polynomial basis for arbitrary mutually dependent probability measure and a sparsity enhancement rotation procedure. This sparsity-enhancing rotation method was initially proposed in our previous work [1] for Gaussian distributions, which may not be feasible for non-Gaussian distributions due to the loss of orthogonality after the rotation. To remedy such difficulties, we developed the new approach to construct orthonormal polynomials for arbitrary mutually dependent (amdP) randomness, ensuring the constructed basis maintains the orthogonality with respect to the density of the rotated random vector, where directly applying the regular polynomial chaos including arbitrary polynomial chaos (aPC) [2] shows limitations due to the assumption of the mutual independence between the components of the random inputs. The developed DSRAR framework leads to accurate recovery of a sparse representation of the target functions. The effectiveness of our method is demonstrated in challenging problems such as PDEs and realistic molecular systems where the underlying density is implicitly represented by a large collection of sample data, as well as systems with explicitly given non-Gaussian probabilistic measures.

Amanda A Howard, Sarah H Murphy, Shady E Ahmed et al.

Physics-informed neural networks and operator networks have shown promise for effectively solving equations modeling physical systems. However, these networks can be difficult or impossible to train accurately for some systems of equations. We present a novel multifidelity framework for stacking physics-informed neural networks and operator networks that facilitates training. We successively build a chain of networks, where the output at one step can act as a low-fidelity input for training the next step, gradually increasing the expressivity of the learned model. The equations imposed at each step of the iterative process can be the same or different (akin to simulated annealing). The iterative (stacking) nature of the proposed method allows us to progressively learn features of a solution that are hard to learn directly. Through benchmark problems including a nonlinear pendulum, the wave equation, and the viscous Burgers equation, we show how stacking can be used to improve the accuracy and reduce the required size of physics-informed neural networks and operator networks.

Shady E. Ahmed, Panos Stinis

Projection-based reduced order models (PROMs) have shown promise in representing the behavior of multiscale systems using a small set of generalized (or latent) variables. Despite their success, PROMs can be susceptible to inaccuracies, even instabilities, due to the improper accounting of the interaction between the resolved and unresolved scales of the multiscale system (known as the closure problem). In the current work, we interpret closure as a multifidelity problem and use a multifidelity deep operator network (DeepONet) framework to address it. In addition, to enhance the stability and accuracy of the multifidelity-based closure, we employ the recently developed "in-the-loop" training approach from the literature on coupling physics and machine learning models. The resulting approach is tested on shock advection for the one-dimensional viscous Burgers equation and vortex merging using the two-dimensional Navier-Stokes equations. The numerical experiments show significant improvement of the predictive ability of the closure-corrected PROM over the un-corrected one both in the interpolative and the extrapolative regimes.

Amanda Howard, Yucheng Fu, Panos Stinis

We introduce a novel continual learning method based on multifidelity deep neural networks. This method learns the correlation between the output of previously trained models and the desired output of the model on the current training dataset, limiting catastrophic forgetting. On its own the multifidelity continual learning method shows robust results that limit forgetting across several datasets. Additionally, we show that the multifidelity method can be combined with existing continual learning methods, including replay and memory aware synapses, to further limit catastrophic forgetting. The proposed continual learning method is especially suited for physical problems where the data satisfy the same physical laws on each domain, or for physics-informed neural networks, because in these cases we expect there to be a strong correlation between the output of the previous model and the model on the current training domain.

Jing Li, Panos Stinis

In many time-dependent problems of practical interest the parameters and/or initial conditions entering the equations describing the evolution of the various quantities exhibit uncertainty. One way to address the problem of how this uncertainty impacts the solution is to expand the solution using polynomial chaos expansions and obtain a system of differential equations for the evolution of the expansion coefficients. We present an application of the Mori-Zwanzig (MZ) formalism to the problem of constructing reduced models of such systems of differential equations. In particular, we construct reduced models for a subset of the polynomial chaos expansion coefficients that are needed for a full description of the uncertainty caused by uncertain parameters or initial conditions. Even though the MZ formalism is exact, its straightforward application to the problem of constructing reduced models for estimating uncertainty involves the computation of memory terms whose cost can become prohibitively expensive. For those cases, we present a Markovian reformulation of the MZ formalism which can lead to approximations that can alleviate some of the computational expense while retaining an accuracy advantage over reduced models that discard the memory altogether. Our results support the conclusion that successful reduced models need to include memory effects.

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.

Wenqian Chen, Panos Stinis

Physics-informed neural networks have emerged as an alternative method for solving partial differential equations. However, for complex problems, the training of such networks can still require high-fidelity data which can be expensive to generate. To reduce or even eliminate the dependency on high-fidelity data, we propose a novel multi-fidelity architecture which is based on a feature space shared by the low- and high-fidelity solutions. In the feature space, the projections of the low-fidelity and high-fidelity solutions are adjacent by constraining their relative distance. The feature space is represented with an encoder and its mapping to the original solution space is effected through a decoder. The proposed multi-fidelity approach is validated on forward and inverse problems for steady and unsteady problems described by partial differential equations.

Shady E. Ahmed, Hui Wan, Saad Qadeer et al.

Toward the goal of using Scientific Machine Learning (SciML) emulators to improve the numerical representation of aerosol processes in global atmospheric models, we explore the emulation of aerosol microphysics processes under cloud-free conditions in the 4-mode Modal Aerosol Module (MAM4) within the Energy Exascale Earth System Model version 2 (E3SMv2). To develop an in-depth understanding of the challenges and opportunities in applying SciML to aerosol processes, we begin with a simple feedforward neural network architecture that has been used in earlier studies, but we systematically examine key emulator design choices, including architecture complexity and variable normalization, while closely monitoring training convergence behavior. Our results show that optimization convergence, scaling strategy, and network complexity strongly influence emulation accuracy. When effective scaling is applied and convergence is achieved, the relatively simple architecture, used together with a moderate network size, can reproduce key features of the microphysics-induced aerosol concentration changes with promising accuracy. These findings provide practical clues for the next stages of emulator development; they also provide general insights that are likely applicable to the emulation of other aerosol processes, as well as other atmospheric physics involving multi-scale variability.

Saad Qadeer, Andrew Engel, Amanda Howard et al.

Despite their immense promise in performing a variety of learning tasks, a theoretical understanding of the limitations of Deep Neural Networks (DNNs) has so far eluded practitioners. This is partly due to the inability to determine the closed forms of the learned functions, making it harder to study their generalization properties on unseen datasets. Recent work has shown that randomly initialized DNNs in the infinite width limit converge to kernel machines relying on a Neural Tangent Kernel (NTK) with known closed form. These results suggest, and experimental evidence corroborates, that empirical kernel machines can also act as surrogates for finite width DNNs. The high computational cost of assembling the full NTK, however, makes this approach infeasible in practice, motivating the need for low-cost approximations. In the current work, we study the performance of the Conjugate Kernel (CK), an efficient approximation to the NTK that has been observed to yield fairly similar results. For the regression problem of smooth functions and logistic regression classification, we show that the CK performance is only marginally worse than that of the NTK and, in certain cases, is shown to be superior. In particular, we establish bounds for the relative test losses, verify them with numerical tests, and identify the regularity of the kernel as the key determinant of performance. In addition to providing a theoretical grounding for using CKs instead of NTKs, our framework suggests a recipe for improving DNN accuracy inexpensively. We present a demonstration of this on the foundation model GPT-2 by comparing its performance on a classification task using a conventional approach and our prescription. We also show how our approach can be used to improve physics-informed operator network training for regression tasks as well as convolutional neural network training for vision classification tasks.

Jacob Price, Panos Stinis

In recent work we have developed a renormalization framework for stabilizing reduced order models for time-dependent partial differential equations. We have applied this framework to the open problem of finite-time singularity formation (blow-up) for the 3D Euler equations of incompressible fluid flow. The renormalized coefficients in the reduced order models decay algebraically with time and resolution. Our results for the behavior of the solutions are consistent with the formation of a finite-time singularity.

Amit Harlev, Andrew Engel, Panos Stinis et al.

Understanding feature representation for deep neural networks (DNNs) remains an open question within the general field of explainable AI. We use principal component analysis (PCA) to study the performance of a k-nearest neighbors classifier (k-NN), nearest class-centers classifier (NCC), and support vector machines on the learned layer-wise representations of a ResNet-18 trained on CIFAR-10. We show that in certain layers, as little as 20% of the intermediate feature-space variance is necessary for high-accuracy classification and that across all layers, the first ~100 PCs completely determine the performance of the k-NN and NCC classifiers. We relate our findings to neural collapse and provide partial evidence for the related phenomenon of intermediate neural collapse. Our preliminary work provides three distinct yet interpretable surrogate models for feature representation with an affine linear model the best performing. We also show that leveraging several surrogate models affords us a clever method to estimate where neural collapse may initially occur within the DNN.

Wenqian Chen, Yucheng Fu, Michael Penwarden et al.

Learning solution operators for systems with complex, varying geometries and parametric physical settings is a central challenge in scientific machine learning. In many-query regimes such as design optimization, control and inverse problems, surrogate modeling must generalize across geometries while allowing flexible evaluation at arbitrary spatial locations. In this work, we propose Arbitrary Geometry-encoded Transformer (ArGEnT), a geometry-aware attention-based architecture for operator learning on arbitrary domains. ArGEnT employs Transformer attention mechanisms to encode geometric information directly from point-cloud representations with three variants-self-attention, cross-attention, and hybrid-attention-that incorporates different strategies for incorporating geometric features. By integrating ArGEnT into DeepONet as the trunk network, we develop a surrogate modeling framework capable of learning operator mappings that depend on both geometric and non-geometric inputs without the need to explicitly parametrize geometry as a branch network input. Evaluation on benchmark problems spanning fluid dynamics, solid mechanics and electrochemical systems, we demonstrate significantly improved prediction accuracy and generalization performance compared with the standard DeepONet and other existing geometry-aware saurrogates. In particular, the cross-attention transformer variant enables accurate geometry-conditioned predictions with reduced reliance on signed distance functions. By combining flexible geometry encoding with operator-learning capabilities, ArGEnT provides a scalable surrogate modeling framework for optimization, uncertainty quantification, and data-driven modeling of complex physical systems.

Panos Stinis

We propose a novel approach which exploits chaos to enhance classification accuracy. Specifically, the available data that need to be classified are treated as vectors that are first lifted into a higher-dimensional space and then used as initial conditions for the evolution of a chaotic dynamical system for a prescribed temporal interval. The evolved state of the dynamical system is then fed to a trainable softmax classifier which outputs the probabilities of the various classes. As proof-of-concept, we use samples of randomly perturbed orthogonal vectors of moderate dimension (2 to 20), with a corresponding number of classes equal to the vector dimension, and show how our approach can both significantly accelerate the training process and improve the classification accuracy compared to a standard softmax classifier which operates on the original vectors, as well as a softmax classifier which only lifts the vectors to a higher-dimensional space without evolving them. We also provide an explanation for the improved performance of the chaos-enhanced classifier.

Bruno Jacob, Amanda A. Howard, Panos Stinis

Quantum algorithms for partial differential equations (PDEs) face severe practical constraints on near-term hardware: limited qubit counts restrict spatial resolution to coarse grids, while circuit depth limitations prevent accurate long-time integration. These hardware bottlenecks confine quantum PDE solvers to low-fidelity regimes despite their theoretical potential for computational speedup. We introduce a multifidelity learning framework that corrects coarse quantum solutions to high-fidelity accuracy using sparse classical training data, facilitating the path toward practical quantum utility for scientific computing. The approach trains a low-fidelity surrogate on abundant quantum solver outputs, then learns correction mappings through a multifidelity neural architecture that balances linear and nonlinear transformations. Demonstrated on benchmark nonlinear PDEs including viscous Burgers equation and incompressible Navier-Stokes flows via quantum lattice Boltzmann methods, the framework successfully corrects coarse quantum predictions and achieves temporal extrapolation well beyond the classical training window. This strategy illustrates how one can reduce expensive high-fidelity simulation requirements while producing predictions that are competitive with classical accuracy. By bridging the gap between hardware-limited quantum simulations and application requirements, this work establishes a pathway for extracting computational value from current quantum devices in real-world scientific applications, advancing both algorithm development and practical deployment of near-term quantum computing for computational physics.

Wenqian Chen, Amanda Howard, Panos Stinis

Physics-informed deep learning has emerged as a promising framework for solving partial differential equations (PDEs). Nevertheless, training these models on complex problems remains challenging, often leading to limited accuracy and efficiency. In this work, we introduce a hybrid adaptive sampling and weighting method to enhance the performance of physics-informed neural networks (PINNs). The adaptive sampling component identifies training points in regions where the solution exhibits rapid variation, while the adaptive weighting component balances the convergence rate across training points. Numerical experiments show that applying only adaptive sampling or only adaptive weighting is insufficient to consistently achieve accurate predictions, particularly when training points are scarce. Since each method emphasizes different aspects of the solution, their effectiveness is problem dependent. By combining both strategies, the proposed framework consistently improves prediction accuracy and training efficiency, offering a more robust approach for solving PDEs with PINNs.

Erdi Kara, Panos Stinis

We present a hybrid framework that couples finite element methods (FEM) with physics-informed DeepONet to model fluid transport in porous media from sharp, localized Gaussian sources. The governing system consists of a steady-state Darcy flow equation and a time-dependent convection-diffusion equation. Our approach solves the Darcy system using FEM and transfers the resulting velocity field to a physics-informed DeepONet, which learns the mapping from source functions to solute concentration profiles. This modular strategy preserves FEM-level accuracy in the flow field while enabling fast inference for transport dynamics. To handle steep gradients induced by sharp sources, we introduce an adaptive sampling strategy for trunk collocation points. Numerical experiments demonstrate that our method is in good agreement with the reference solutions while offering orders of magnitude speedups over traditional solvers, making it suitable for practical applications in relevant scenarios. Implementation of our proposed method is available at https://github.com/erkara/fem-pi-deeponet.

Bruno Jacob, Amanda A. Howard, Panos Stinis

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) in scientific computing. While PINNs typically use multilayer perceptrons (MLPs) as their underlying architecture, recent advancements have explored alternative neural network structures. One such innovation is the Kolmogorov-Arnold Network (KAN), which has demonstrated benefits over traditional MLPs, including faster neural scaling and better interpretability. The application of KANs to physics-informed learning has led to the development of Physics-Informed KANs (PIKANs), enabling the use of KANs to solve PDEs. However, despite their advantages, KANs often suffer from slower training speeds, particularly in higher-dimensional problems where the number of collocation points grows exponentially with the dimensionality of the system. To address this challenge, we introduce Separable Physics-Informed Kolmogorov-Arnold Networks (SPIKANs). This novel architecture applies the principle of separation of variables to PIKANs, decomposing the problem such that each dimension is handled by an individual KAN. This approach drastically reduces the computational complexity of training without sacrificing accuracy, facilitating their application to higher-dimensional PDEs. Through a series of benchmark problems, we demonstrate the effectiveness of SPIKANs, showcasing their superior scalability and performance compared to PIKANs and highlighting their potential for solving complex, high-dimensional PDEs in scientific computing.

Amanda A. Howard, Bruno Jacob, Panos Stinis

We develop a method for multifidelity Kolmogorov-Arnold networks (KANs), which use a low-fidelity model along with a small amount of high-fidelity data to train a model for the high-fidelity data accurately. Multifidelity KANs (MFKANs) reduce the amount of expensive high-fidelity data needed to accurately train a KAN by exploiting the correlations between the low- and high-fidelity data to give accurate and robust predictions in the absence of a large high-fidelity dataset. In addition, we show that multifidelity KANs can be used to increase the accuracy of physics-informed KANs (PIKANs), without the use of training data.

Sifan Wang, Shyam Sankaran, Xiantao Fan et al.

Turbulent fluid flows are among the most computationally demanding problems in science, requiring enormous computational resources that become prohibitive at high flow speeds. Physics-informed neural networks (PINNs) represent a radically different approach that trains neural networks directly from physical equations rather than data, offering the potential for continuous, mesh-free solutions. Here we show that appropriately designed PINNs can successfully simulate fully turbulent flows in both two and three dimensions, directly learning solutions to the fundamental fluid equations without traditional computational grids or training data. Our approach combines several algorithmic innovations including adaptive network architectures, causal training, and advanced optimization methods to overcome the inherent challenges of learning chaotic dynamics. Through rigorous validation on challenging turbulence problems, we demonstrate that PINNs accurately reproduce key flow statistics including energy spectra, kinetic energy, enstrophy, and Reynolds stresses. Our results demonstrate that neural equation solvers can handle complex chaotic systems, opening new possibilities for continuous turbulence modeling that transcends traditional computational limitations.

Alexander Heinlein, Amanda A. Howard, Damien Beecroft et al.

Multiscale problems are challenging for neural network-based discretizations of differential equations, such as physics-informed neural networks (PINNs). This can be (partly) attributed to the so-called spectral bias of neural networks. To improve the performance of PINNs for time-dependent problems, a combination of multifidelity stacking PINNs and domain decomposition-based finite basis PINNs is employed. In particular, to learn the high-fidelity part of the multifidelity model, a domain decomposition in time is employed. The performance is investigated for a pendulum and a two-frequency problem as well as the Allen-Cahn equation. It can be observed that the domain decomposition approach clearly improves the PINN and stacking PINN approaches. Finally, it is demonstrated that the FBPINN approach can be extended to multifidelity physics-informed deep operator networks.

Amirhossein Mollaali, Christian Bolivar Moya, Amanda A. Howard et al.

This paper explores uncertainty quantification (UQ) methods in the context of Kolmogorov-Arnold Networks (KANs). We apply an ensemble approach to KANs to obtain a heuristic measure of UQ, enhancing interpretability and robustness in modeling complex functions. Building on this, we introduce Conformalized-KANs, which integrate conformal prediction, a distribution-free UQ technique, with KAN ensembles to generate calibrated prediction intervals with guaranteed coverage. Extensive numerical experiments are conducted to evaluate the effectiveness of these methods, focusing particularly on the robustness and accuracy of the prediction intervals under various hyperparameter settings. We show that the conformal KAN predictions can be applied to recent extensions of KANs, including Finite Basis KANs (FBKANs) and multifideilty KANs (MFKANs). The results demonstrate the potential of our approaches to improve the reliability and applicability of KANs in scientific machine learning.

Bruno Jacob, Ashish S. Nair, Amanda A. Howard et al.

Physics-informed neural networks (PINNs) have demonstrated promise as a framework for solving forward and inverse problems involving partial differential equations. Despite recent progress in the field, it remains challenging to quantify uncertainty in these networks. While techniques such as Bayesian PINNs (B-PINNs) provide a principled approach to capturing epistemic uncertainty through Bayesian inference, they can be computationally expensive for large-scale applications. In this work, we propose Epistemic Physics-Informed Neural Networks (E-PINNs), a framework that uses a small network, the epinet, to efficiently quantify epistemic uncertainty in PINNs. The proposed approach works as an add-on to existing, pre-trained PINNs with a small computational overhead. We demonstrate the applicability of the proposed framework in various test cases and compare the results with B-PINNs using Hamiltonian Monte Carlo (HMC) posterior estimation and dropout-equipped PINNs (Dropout-PINNs). In our experiments, E-PINNs achieve calibrated coverage with competitive sharpness at substantially lower cost. We demonstrate that when B-PINNs produce narrower bands, they under-cover in our tests. E-PINNs also show better calibration than Dropout-PINNs in these examples, indicating a favorable accuracy-efficiency trade-off.

Emily Williams, Amanda Howard, Brek Meuris et al.

Physics-informed deep operator networks (DeepONets) have emerged as a promising approach toward numerically approximating the solution of partial differential equations (PDEs). In this work, we aim to develop further understanding of what is being learned by physics-informed DeepONets by assessing the universality of the extracted basis functions and demonstrating their potential toward model reduction with spectral methods. Results provide clarity about measuring the performance of a physics-informed DeepONet through the decays of singular values and expansion coefficients. In addition, we propose a transfer learning approach for improving training for physics-informed DeepONets between parameters of the same PDE as well as across different, but related, PDEs where these models struggle to train well. This approach results in significant error reduction and learned basis functions that are more effective in representing the solution of a PDE.

Pietro Zanotta, Panos Stinis, Ján Drgoňa

Certifying the Region of Attraction (ROA) for high-dimensional nonlinear dynamical systems remains a severe computational bottleneck. Traditional deterministic verification methods, such as Sum-of-Squares (SOS) programming and Satisfiability Modulo Theories (SMT), provide hard guarantees but suffer from the curse of dimensionality, typically failing to scale beyond 20 dimensions. To overcome these limitations, we propose SCORE, a statistical certification framework that shifts from seeking deterministic guarantees to bounding the worst-case safety violation with high statistical confidence. By integrating Projected Stochastic Gradient Langevin Dynamics (PSGLD) with Extreme Value Theory (EVT), we frame ROA certification as a constrained extreme-value estimation problem on the sublevel set boundary. We theoretically demonstrate that modeling the optimization process as a stochastic diffusion on a compact manifold places the local maxima of the Lyapunov derivative into the Weibull maximum domain of attraction. Since the Weibull domain features a finite right endpoint, we can compute a rigorous statistical upper bound on the global maximum of the Lyapunov derivative. Numerical experiments validate that our EVT-based approach achieves certification tightness competitive to exact SOS programming on a 2D Van der Pol benchmark. Furthermore, we demonstrate unprecedented scalability by successfully certifying a dense, unstructured 500-dimensional ODE system up to a confidence level of 99.99\%, effectively bypassing the severe combinatorial constraints that limit existing formal verification pipelines.

Saad Qadeer, Panos Stinis

Physics-informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not obvious, with the result that they typically yield moderately accurate solutions. In this article, we propose a method for improving the accuracy of PINNs by coupling them with a post-processing step that seeks the best approximation in a function space associated with the network. We find that our method yields errors four to five orders of magnitude lower than those of the parent PINNs across architectures and dimensions. Moreover, we can reuse the basis functions for the linear space in more complex settings, such as time-dependent and nonlinear problems, allowing for transfer learning. Out approach also provides a residual-based metric that allows us to optimally choose the number of basis functions employed.

Zhi-Feng Wei, Wenqian Chen, Panos Stinis

Operator learning has emerged as a promising tool for accelerating the solution of partial differential equations (PDEs). The Deep Operator Networks (DeepONets) represent a pioneering framework in this area: the "vanilla" DeepONet is valued for its simplicity and efficiency, while the modified DeepONet achieves higher accuracy at the cost of increased training time. In this work, we propose a series of Transformer-inspired DeepONet variants that introduce bidirectional cross-conditioning between the branch and trunk networks in DeepONet. Query-point information is injected into the branch network and input-function information into the trunk network, enabling dynamic dependencies while preserving the simplicity and efficiency of the "vanilla" DeepONet in a non-intrusive manner. Experiments on four PDE benchmarks -- advection, diffusion-reaction, Burgers', and Korteweg-de Vries equations -- show that for each case, there exists a variant that matches or surpasses the accuracy of the modified DeepONet while offering improved training efficiency. Moreover, the best-performing variant for each equation aligns naturally with the equation's underlying characteristics, suggesting that the effectiveness of cross-conditioning depends on the characteristics of the equation and its underlying physics. To ensure robustness, we validate the effectiveness of our variants through a range of rigorous statistical analyses, among them the Wilcoxon Two One-Sided Test, Glass's Delta, and Spearman's rank correlation.

Saad Qadeer, Panos Stinis, Hui. Wan

A long-standing obstacle in the use of machine-learnt surrogates with larger PDE systems is the onset of instabilities when solved numerically. Efforts towards ameliorating these have mostly concentrated on improving the accuracy of the surrogates or imbuing them with additional structure, and have garnered limited success. In this article, we study a prototype problem and draw insights that can help with more complex systems. In particular, we focus on a viscous Burgers'-ML system and, after identifying the cause of the instabilities, prescribe strategies to stabilize the coupled system. To improve the accuracy of the stabilized system, we next explore methods based on the Mori--Zwanzig formalism.

Amanda A. Howard, Bruno Jacob, Sarah Helfert et al.

Kolmogorov-Arnold networks (KANs) have attracted attention recently as an alternative to multilayer perceptrons (MLPs) for scientific machine learning. However, KANs can be expensive to train, even for relatively small networks. Inspired by finite basis physics-informed neural networks (FBPINNs), in this work, we develop a domain decomposition method for KANs that allows for several small KANs to be trained in parallel to give accurate solutions for multiscale problems. We show that finite basis KANs (FBKANs) can provide accurate results with noisy data and for physics-informed training.

Wenqian Chen, Amanda A. Howard, Panos Stinis

Physics-informed deep learning has emerged as a promising alternative for solving partial differential equations. However, for complex problems, training these networks can still be challenging, often resulting in unsatisfactory accuracy and efficiency. In this work, we demonstrate that the failure of plain physics-informed neural networks arises from the significant discrepancy in the convergence rate of residuals at different training points, where the slowest convergence rate dominates the overall solution convergence. Based on these observations, we propose a pointwise adaptive weighting method that balances the residual decay rate across different training points. The performance of our proposed adaptive weighting method is compared with current state-of-the-art adaptive weighting methods on benchmark problems for both physics-informed neural networks and physics-informed deep operator networks. Through extensive numerical results we demonstrate that our proposed approach of balanced residual decay rates offers several advantages, including bounded weights, high prediction accuracy, fast convergence rate, low training uncertainty, low computational cost, and ease of hyperparameter tuning.

Wenqian Chen, Peiyuan Gao, Panos Stinis

The Ornstein-Zernike (OZ) equation is the fundamental equation for pair correlation function computations in the modern integral equation theory for liquids. In this work, machine learning models, notably physics-informed neural networks and physics-informed neural operator networks, are explored to solve the OZ equation. The physics-informed machine learning models demonstrate great accuracy and high efficiency in solving the forward and inverse OZ problems of various bulk fluids. The results highlight the significant potential of physics-informed machine learning for applications in thermodynamic state theory.

Wenqian Chen, Yucheng Fu, Panos Stinis

In this paper, we present a physics-informed neural network (PINN) approach for predicting the performance of an all-vanadium redox flow battery, with its physics constraints enforced by a two-dimensional (2D) mathematical model. The 2D model, which includes 6 governing equations and 24 boundary conditions, provides a detailed representation of the electrochemical reactions, mass transport and hydrodynamics occurring inside the redox flow battery. To solve the 2D model with the PINN approach, a composite neural network is employed to approximate species concentration and potentials; the input and output are normalized according to prior knowledge of the battery system; the governing equations and boundary conditions are first scaled to an order of magnitude around 1, and then further balanced with a self-weighting method. Our numerical results show that the PINN is able to predict cell voltage correctly, but the prediction of potentials shows a constant-like shift. To fix the shift, the PINN is enhanced by further constrains derived from the current collector boundary. Finally, we show that the enhanced PINN can be even further improved if a small number of labeled data is available.

Brek Meuris, Saad Qadeer, Panos Stinis

Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet), to identify candidate functions on which to expand the solution of PDEs. We have devised an approach which uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions which have the following properties: i) they constitute a basis, 2) they are orthonormal, and 3) they are hierarchical i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs.

Kookjin Lee, Nathaniel Trask, Panos Stinis

Discovery of dynamical systems from data forms the foundation for data-driven modeling and recently, structure-preserving geometric perspectives have been shown to provide improved forecasting, stability, and physical realizability guarantees. We present here a unification of the Sparse Identification of Nonlinear Dynamics (SINDy) formalism with neural ordinary differential equations. The resulting framework allows learning of both "black-box" dynamics and learning of structure preserving bracket formalisms for both reversible and irreversible dynamics. We present a suite of benchmarks demonstrating effectiveness and structure preservation, including for chaotic systems.

Kookjin Lee, Nathaniel A. Trask, Panos Stinis

Forecasting of time-series data requires imposition of inductive biases to obtain predictive extrapolation, and recent works have imposed Hamiltonian/Lagrangian form to preserve structure for systems with reversible dynamics. In this work we present a novel parameterization of dissipative brackets from metriplectic dynamical systems appropriate for learning irreversible dynamics with unknown a priori model form. The process learns generalized Casimirs for energy and entropy guaranteed to be conserved and nondecreasing, respectively. Furthermore, for the case of added thermal noise, we guarantee exact preservation of a fluctuation-dissipation theorem, ensuring thermodynamic consistency. We provide benchmarks for dissipative systems demonstrating learned dynamics are more robust and generalize better than either "black-box" or penalty-based approaches.

QiZhi He, Panos Stinis, Alexandre Tartakovsky

In this paper, we present a physics-constrained deep neural network (PCDNN) method for parameter estimation in the zero-dimensional (0D) model of the vanadium redox flow battery (VRFB). In this approach, we use deep neural networks (DNNs) to approximate the model parameters as functions of the operating conditions. This method allows the integration of the VRFB computational models as the physical constraints in the parameter learning process, leading to enhanced accuracy of parameter estimation and cell voltage prediction. Using an experimental dataset, we demonstrate that the PCDNN method can estimate model parameters for a range of operating conditions and improve the 0D model prediction of voltage compared to the 0D model prediction with constant operation-condition-independent parameters estimated with traditional inverse methods. We also demonstrate that the PCDNN approach has an improved generalization ability for estimating parameter values for operating conditions not used in the DNN training.

Panos Stinis

We assume that we are given a time series of data from a dynamical system and our task is to learn the flow map of the dynamical system. We present a collection of results on how to enforce constraints coming from the dynamical system in order to accelerate the training of deep neural networks to represent the flow map of the system as well as increase their predictive ability. In particular, we provide ways to enforce constraints during training for all three major modes of learning, namely supervised, unsupervised and reinforcement learning. In general, the dynamic constraints need to include terms which are analogous to memory terms in model reduction formalisms. Such memory terms act as a restoring force which corrects the errors committed by the learned flow map during prediction. For supervised learning, the constraints are added to the objective function. For the case of unsupervised learning, in particular generative adversarial networks, the constraints are introduced by augmenting the input of the discriminator. Finally, for the case of reinforcement learning and in particular actor-critic methods, the constraints are added to the reward function. In addition, for the reinforcement learning case, we present a novel approach based on homotopy of the action-value function in order to stabilize and accelerate training. We use numerical results for the Lorenz system to illustrate the various constructions.

Ramakrishna Tipireddy, Paris Perdikaris, Panos Stinis et al.

We investigate the use of discrete and continuous versions of physics-informed neural network methods for learning unknown dynamics or constitutive relations of a dynamical system. For the case of unknown dynamics, we represent all the dynamics with a deep neural network (DNN). When the dynamics of the system are known up to the specification of constitutive relations (that can depend on the state of the system), we represent these constitutive relations with a DNN. The discrete versions combine classical multistep discretization methods for dynamical systems with neural network based machine learning methods. On the other hand, the continuous versions utilize deep neural networks to minimize the residual function for the continuous governing equations. We use the case of a fedbatch bioreactor system to study the effectiveness of these approaches and discuss conditions for their applicability. Our results indicate that the accuracy of the trained neural network models is much higher for the cases where we only have to learn a constitutive relation instead of the whole dynamics. This finding corroborates the well-known fact from scientific computing that building as much structural information is available into an algorithm can enhance its efficiency and/or accuracy.

Nathan O. Hodas, Panos Stinis

As deep neural networks grow in size, from thousands to millions to billions of weights, the performance of those networks becomes limited by our ability to accurately train them. A common naive question arises: if we have a system with billions of degrees of freedom, don't we also need billions of samples to train it? Of course, the success of deep learning indicates that reliable models can be learned with reasonable amounts of data. Similar questions arise in protein folding, spin glasses and biological neural networks. With effectively infinite potential folding/spin/wiring configurations, how does the system find the precise arrangement that leads to useful and robust results? Simple sampling of the possible configurations until an optimal one is reached is not a viable option even if one waited for the age of the universe. On the contrary, there appears to be a mechanism in the above phenomena that forces them to achieve configurations that live on a low-dimensional manifold, avoiding the curse of dimensionality. In the current work we use the concept of mutual information between successive layers of a deep neural network to elucidate this mechanism and suggest possible ways of exploiting it to accelerate training. We show that adding structure to the neural network that enforces higher mutual information between layers speeds training and leads to more accurate results. High mutual information between layers implies that the effective number of free parameters is exponentially smaller than the raw number of tunable weights.

Panos Stinis, Tobias Hagge, Alexandre M. Tartakovsky et al.

We suggest ways to enforce given constraints in the output of a Generative Adversarial Network (GAN) generator both for interpolation and extrapolation (prediction). For the case of dynamical systems, given a time series, we wish to train GAN generators that can be used to predict trajectories starting from a given initial condition. In this setting, the constraints can be in algebraic and/or differential form. Even though we are predominantly interested in the case of extrapolation, we will see that the tasks of interpolation and extrapolation are related. However, they need to be treated differently. For the case of interpolation, the incorporation of constraints is built into the training of the GAN. The incorporation of the constraints respects the primary game-theoretic setup of a GAN so it can be combined with existing algorithms. However, it can exacerbate the problem of instability during training that is well-known for GANs. We suggest adding small noise to the constraints as a simple remedy that has performed well in our numerical experiments. The case of extrapolation (prediction) is more involved. During training, the GAN generator learns to interpolate a noisy version of the data and we enforce the constraints. This approach has connections with model reduction that we can utilize to improve the efficiency and accuracy of the training. Depending on the form of the constraints, we may enforce them also during prediction through a projection step. We provide examples of linear and nonlinear systems of differential equations to illustrate the various constructions.

Tobias Hagge, Panos Stinis, Enoch Yeung et al.

We solve a system of ordinary differential equations with an unknown functional form of a sink (reaction rate) term. We assume that the measurements (time series) of state variables are partially available, and we use recurrent neural network to "learn" the reaction rate from this data. This is achieved by including a discretized ordinary differential equations as part of a recurrent neural network training problem. We extend TensorFlow's recurrent neural network architecture to create a simple but scalable and effective solver for the unknown functions, and apply it to a fedbatch bioreactor simulation problem. Use of techniques from recent deep learning literature enables training of functions with behavior manifesting over thousands of time steps. Our networks are structurally similar to recurrent neural networks, but differences in design and function require modifications to the conventional wisdom about training such networks.

Ramakrishna Tipireddy, Panos Stinis, Alexandre Tartakovsky

We present a novel uncertainty quantification approach for high-dimensional stochastic partial differential equations that reduces the computational cost of polynomial chaos methods by decomposing the computational domain into non-overlapping subdomains and adapting the stochastic basis in each subdomain so the local solution has a lower dimensional random space representation. The local solutions are coupled using the Neumann-Neumann algorithm, where we first estimate the interface solution then evaluate the interior solution in each subdomain using the interface solution as a boundary condition. The interior solutions in each subdomain are computed independently of each other, which reduces the operation count from $O(N^α)$ to $O(M^α),$ where $N$ is the total number of degrees of freedom, $M$ is the number of degrees of freedom in each subdomain, and the exponent $α>1$ depends on the uncertainty quantification method used. In addition, the localized nature of solutions makes the proposed approach highly parallelizable. We illustrate the accuracy and efficiency of the approach for linear and nonlinear differential equations with random coefficients.

Jacob Price, Panos Stinis

We examine the challenging problem of constructing reduced models for the long time prediction of systems where there is no timescale separation between the resolved and unresolved variables. In previous work we focused on the case where there was only transfer of activity (e.g. energy, mass) from the resolved to the unresolved variables. Here we investigate the much more difficult case where there is two-way transfer of activity between the resolved and unresolved variables. Like in the case of activity drain out of the resolved variables, even if one starts with an exact formalism, like the Mori-Zwanzig (MZ) formalism, the constructed reduced models can become unstable. We show how to remedy this situation by using dynamic information from the full system to renormalize the MZ reduced models. In addition to being stabilized, the renormalized models can be accurate for very long times. We use the Korteweg-de Vries equation to illustrate the approach. The coefficients of the renormalized models exhibit rich structure, including algebraic time dependence and incomplete similarity.