arXiv subjects

Panos Stinis

Publications and source records attributed to Panos Stinis.

At least 19 recordsLinked to original sources

arXiv2026

In-Context Learning for Multi-Fidelity Multiscale Model Closure and Cross-Operator Out-Of-Distribution Generalization

Multi-fidelity learning uses low-fidelity information (coarse-scale PDE models) together with limited high-fidelity data (fine-scale solutions) to obtain high-accuracy PDE solutions. It is a data-driven approach to downscale reduced-order models to solve PDEs efficiently and accurately. Standard multi-fidelity learning focuses on one operator or PDE; given a new PDE, retraining using additional coarse-fine solution pairs may be required, which is costly. In this work, we propose an in-context multi-fidelity learning framework for learning multiple multi-fidelity operators with one shared model. The model is trained using paired low- and high-fidelity examples as demonstrations from multiple operators, together with a new coarse-scale solution query, to predict the corresponding fine-scale solution. Once trained, given a new PDE operator, a few (fewer than five) coarse-fine demonstration pairs, and a low-cost coarse query, the model predicts the corresponding fine-scale solution without fine-tuning. When the target operator appears in the training dataset but has a different prediction goal, such as predicting beyond the training time interval, no additional demonstrations are needed. Instead, the model uses training coarse-fine pairs from that PDE and information from other training operators. Testing then only requires generating a low-cost coarse-scale solution query, without fine-tuning. Notably, our smallest model contains only 255,364 trainable parameters, corresponding to approximately 456 parameters per training operator for 560 operators.

math.NA↗

arXiv2026

SCORE: Statistical Certification of Regions of Attraction via Extreme Value Theory

Certifying the Region of Attraction (ROA) for high-dimensional nonlinear dynamical systems remains a severe computational bottleneck. Traditional deterministic verification methods 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. Under stationary sampling and a regular local-geometry condition around the global maximum, we show that the Lyapunov derivative belongs to the Weibull maximum domain of attraction. Its finite right endpoint enables statistical estimation of the global maximum of the Lyapunov derivative and construction of an upper confidence bound, conditional on the sampling and inference assumptions. Numerical experiments validate that our EVT-based approach achieves certification tightness competitive to exact Sum of Squares programming on a 2D Van der Pol benchmark. Furthermore, we demonstrate strong scalability by successfully applying the statistical verification procedure to a dense, unstructured 500-dimensional ODE system at a nominal confidence level of 99.99\%, effectively bypassing the severe combinatorial constraints that limit existing formal verification pipelines.

eess.SY↗

arXiv2026

Mamba-Assisted Non-Markovian Closure for Reduced-Order Modeling

Reduced-order modeling of high-dimensional dynamical systems is often hindered by closure effects arising from unresolved variables, which can introduce non-Markovian dependence into the resolved dynamics. Motivated by the history-dependent memory term arising in the Mori--Zwanzig formalism, we recast non-Markovian closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework. MAC employs a Mamba-based sequence model to predict the closure from the resolved trajectory and couples the learned closure with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. During training, the selective scan mechanism in Mamba enables efficient parallel sequence processing with linear scaling in sequence length, while autoregressive inference proceeds through recurrent state updates at essentially constant per-step cost. We evaluate MAC on four benchmark systems with complementary characteristics: the viscous Burgers' equation, the chaotic two-scale Lorenz '96 system, the 3-bus DeMarco--Zheng power-grid system, and the dispersive Korteweg--de Vries equation. Across these benchmarks, MAC consistently improves predictive accuracy and long-time rollout stability relative to the comparison models, demonstrating an effective and computationally scalable approach to non-Markovian closure modeling.

cs.LG↗

arXiv2026

Uncertainty-aware Multi-fidelity Closure via Conditional Normalizing Flows

Reduced-order models (ROMs) provide efficient surrogates for complex multiscale systems, but their predictive accuracy is often compromised by truncation errors and the inadequate representation of interactions between resolved and unresolved scales. The missing effect of truncated (unresolved) scales on ROM (resolved) scales is often denoted as the closure problem. In this work, we formulate ROM closure modeling as a multi-fidelity (MF) learning problem and propose an uncertainty-aware MF framework based on conditional normalizing flows to enhance ROM predictive accuracy. The proposed approach learns a probabilistic mapping from low-fidelity (LF) ROM coefficients to high-fidelity (HF) coefficients, thereby improving predictive fidelity while quantifying the uncertainty associated with the learned closure. Two correction strategies are investigated: direct learning, in which HF coefficients are predicted directly from LF inputs, and residual learning, which learns the discrepancy between LF and HF coefficients. The framework is demonstrated on a double shear layer and vortex merging problems governed by the two-dimensional NavierStokes equations. Results show that both correction strategies improve ROM accuracy over uncorrected ROM, with residual learning achieving consistently better performance than direct learning. Moreover, the two proposed deep generative model-based strategies provide uncertainty quantification for the corrected ROM coefficients, which is critical for assessing prediction confidence and supporting the reliable use of ROMs in practical applications.

cs.LG↗

arXiv2026

ArGEnT: Arbitrary Geometry-encoded Transformer for Operator Learning

Learning solution operators on arbitrary geometries remains a central challenge in scientific machine learning, especially for many-query simulation, physics-informed learning, and evolving geometries requiring accurate, geometry-aware predictions at arbitrary spatial locations. Existing operator-learning methods often rely on structured discretizations, explicit geometry parameterizations, or point-cloud formulations that couple geometric representation with solution-query sampling, limiting flexibility on irregular and non-parameterized domains. We propose the Arbitrary Geometry-encoded Transformer (ArGEnT), a geometry-conditioned attention framework that decouples geometry encoding from query-point evaluation. We develop three variants: self-attention, cross-attention, and hybrid-attention. ArGEnT can be used independently or integrated with neural operators to incorporate non-geometric physical inputs. In the cross-attention variant, geometry is represented by an independently sampled point cloud used to construct keys and values, while arbitrary solution-query points construct queries. This design enables mesh-independent field prediction, reduces sensitivity to query-point distribution, and allows compact geometric representations to condition large-scale solution evaluations. Across benchmarks in fluid dynamics, solid mechanics, and electrochemical systems, ArGEnT consistently improves accuracy and generalization over standard DeepONet, point-cloud-based operator learning, and geometry-aware transformer baselines. In several cases, it reduces prediction errors by more than an order of magnitude while requiring substantially lower training cost than transformer-based baselines. These results demonstrate that decoupled geometry-query attention provides an accurate, scalable, and flexible framework for operator learning on arbitrary geometries.

cs.LG↗

arXiv2026

From Classification to Regression: Using a Fruitfly to Solve Equations

We present a novel approach to regression tasks using classification which is motivated by the mechanism used by fruitflies to sense their environment. Specifically, we formulate a general framework for learning nonlinear input-output relationships by replacing complex global surrogate models with a finite library of representative local patterns. Since scientific data often occupy limited and recurring regions of the input space, we generate predictions by measuring similarities between a query and stored patterns, then combining their associated responses through weighted reconstruction. We apply this approach to nonlinear dynamical systems, data-driven regression, and physics-informed learning using suitable embeddings and similarity measures. For dynamical systems, our offline-online workflow extracts patterns from data or governing equations during the offline phase, while online prediction requires only similarity evaluation and response aggregation. This structure helps us reduce computational and memory demands while providing explicit control over the trade-off among accuracy, storage, and inference cost.

cs.LG↗

arXiv2026

An adaptive phase field framework for large-scale interface evolution problems using a strong-form gradient smoothing approach

Multiscale problems with evolving interfaces are ubiquitous in science and engineering. Phase-field models are a powerful tool for simulating interface-dominated phenomena in computational mechanics and materials modeling, but their application to large-scale problems is often constrained by the high computational cost of resolving thin diffuse interfaces over the entire domain. This paper presents an efficient strong-form phase-field solver that couples the Gradient Smoothing Method (GSM) with a hierarchical adaptive and moving structured mesh, enabling automatic localization of resolution within a narrow interfacial region while retaining coarse discretization in bulk domains. A layered refinement design is introduced to preserve locally uniform resolution across the interface, allowing the GSM discretization to maintain overall second-order accuracy despite strong mesh non-uniformity away from the interface. Although GSM incurs a higher per-degree-of-freedom cost than standard finite-difference schemes, the adaptive framework substantially reduces the total number of degrees of freedom, resulting in near-linear computational scaling compared with the quadratic scaling of uniform-grid approaches. Numerical examples based on the Allen-Cahn and Cahn-Hilliard equations demonstrate that the proposed adaptive GSM solver delivers desired accuracy for interface evolution while attaining more favorable computational complexity, O(N), than existing weak-form and strong-form solvers, becoming significantly more efficient for large-scale problems with thin interfaces or a small interfacial area fraction relative to the whole domain.

math.NA↗

arXiv2026

Assessing Emulator Design and Training for Modal Aerosol Microphysics Parameterizations in E3SMv2

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.

physics.ao-ph↗

arXiv2026

Enhancing classification accuracy through chaos

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 and a selection process for the optimal chaotic evolution interval.

cs.LG↗

arXiv2026

SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networks

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.

cs.LG↗

arXiv2026

Improving the accuracy of physics-informed neural networks via last-layer retraining

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. Our approach also provides a residual-based metric that allows us to optimally choose the number of basis functions employed.

math.NA↗

arXiv2025

Bridging quantum and classical computing for partial differential equations through multifidelity machine learning

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.

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

Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems

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.

cs.LG↗

arXiv2025

E-PINNs: Epistemic Physics-Informed Neural Networks

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.

cs.LG↗

arXiv2025

Self-adaptive weighting and sampling for physics-informed neural networks

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.

stat.ML↗

arXiv2025

From LIF to QIF: Toward Differentiable Spiking Neurons for Scientific Machine Learning

Spiking neural networks (SNNs) offer biologically inspired computation but remain underexplored for continuous regression tasks in scientific machine learning. In this work, we introduce and systematically evaluate Quadratic Integrate-and-Fire (QIF) neurons as an alternative to the conventional Leaky Integrate-and-Fire (LIF) model in both directly trained SNNs and ANN-to-SNN conversion frameworks. The QIF neuron exhibits smooth and differentiable spiking dynamics, enabling gradient-based training and stable optimization within architectures such as multilayer perceptrons (MLPs), Deep Operator Networks (DeepONets), and Physics-Informed Neural Networks (PINNs). Across benchmarks on function approximation, operator learning, and partial differential equation (PDE) solving, QIF-based networks yield smoother, more accurate, and more stable predictions than their LIF counterparts, which suffer from discontinuous time-step responses and jagged activation surfaces. These results position the QIF neuron as a computational bridge between spiking and continuous-valued deep learning, advancing the integration of neuroscience-inspired dynamics into physics-informed and operator-learning frameworks.

cs.NE↗

arXiv2025

Stabilizing PDE--ML coupled systems

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.

math.NA↗