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We introduce a novel framework for uncertainty quantification of solution operators associated with stochastic partial differential equations (SPDEs). Although SPDEs play a central role in modeling complex physical systems under…

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc.…

NOMAD (Navigating Optimal Model Application for Datastreams) is an intelligent framework for data enrichment during ingestion that optimizes realtime multiclass classification by dynamically constructing model chains, i.e ,sequences of…

Learning dynamics governed by differential equations is crucial for predicting and controlling the systems in science and engineering. Neural Ordinary Differential Equation (NODE), a deep learning model integrated with differential…

Neural operators aim to approximate the solution operator of a system of differential equations purely from data. They have shown immense success in modeling complex dynamical systems across various domains. However, the occurrence of…

We study a novel alternative approach for the computation of transport coefficients at mesoscales. While standard nonequilibrium molecular dynamics (NEMD) approaches fix the forcing and measure the average induced flux in the system driven…

This paper presents a nonlinear reduced-order modeling (ROM) framework that leverages deep learning and manifold learning to predict compressible flow fields with complex nonlinear features, including shock waves. The proposed DeepManifold…

Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use…

Nonlinear phenomena can be analyzed via linear techniques using operator-theoretic approaches. Data-driven method called the extended dynamic mode decomposition (EDMD) and its variants, which approximate the Koopman operator associated with…

Neural ordinary differential equations (NODE) have garnered significant attention for their design of continuous-depth neural networks and the ability to learn data/feature dynamics. However, for high-dimensional systems, estimating…

We introduce the Gram-Hadamard Density Operator (GHDO), a new deep neural-network architecture that can encode positive semi-definite density operators of exponential rank with polynomial resources. We then show how to embed an…

The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral…

Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based…

Panel data involving longitudinal measurements of the same set of participants taken over multiple time points is common in studies to understand childhood development and disease modeling. Deep hybrid models that marry the predictive power…

Neural ordinary differential equations (NODEs) have been proven useful for learning non-linear dynamics of arbitrary trajectories. However, current NODE methods capture variations across trajectories only via the initial state value or by…

We study the approximation-theoretic implications of mixture-of-experts architectures for operator learning, where the complexity of a single large neural operator is distributed across many small neural operators (NOs), and each input is…

Interfacial dynamics underlie a wide range of phenomena, including phase transitions, microstructure coarsening, pattern formation, and thin-film growth, and are typically described by stiff, time-dependent nonlinear partial differential…

Neural operators have emerged as an efficient paradigm for solving PDEs, overcoming the limitations of traditional numerical methods and significantly improving computational efficiency. However, due to the diversity and complexity of PDE…

We introduce MENO (''Matrix Exponential-based Neural Operator''), a hybrid surrogate modeling framework for efficiently solving stiff systems of ordinary differential equations (ODEs) that exhibit a sparse nonlinear structure. In such…