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Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
In this work, we establish a deformation-based framework for learning solution mappings of PDEs defined on varying domains. The union of functions defined on varying domains can be identified as a metric space according to the deformation,…
We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial…
To speed-up the solution to parametrized differential problems, reduced order models (ROMs) have been developed over the years, including projection-based ROMs such as the reduced-basis (RB) method, deep learning-based ROMs, as well as…
Deep Operator Networks (DeepONets) are among the most prominent frameworks for operator learning, grounded in the universal approximation theorem for operators. However, training DeepONets typically requires significant computational…
Recent advances in machine learning (ML) have opened new possibilities for solving partial differential equations (PDEs), yet robust performance in challenging regimes remains limited. In particular, singularly perturbed differential…
This focused review explores a range of neural operator architectures for approximating solutions to parametric partial differential equations (PDEs), emphasizing high-level concepts and practical implementation strategies. The study covers…
High-performance deep neural network (DNN)-based systems are in high demand in edge environments. Due to its high computational complexity, it is challenging to deploy DNNs on edge devices with strict limitations on computational resources.…
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.…
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…
Neural surrogates for stiff differential-algebraic equations (DAEs) face two barriers: soft-constraint methods leave algebraic residuals that stiffness amplifies into errors, and hard-constraint methods require trajectory data from stiff…
Deep learning has emerged as a compelling framework for scientific and engineering computing, motivating growing interest in neural network-based solvers for partial differential equations (PDEs). Within this landscape, network…
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
Developing the proper representations for simulating high-speed flows with strong shock waves, rarefactions, and contact discontinuities has been a long-standing question in numerical analysis. Herein, we employ neural operators to solve…
Finding parameters in a deep neural network (NN) that fit training data is a nonconvex optimization problem, but a basic first-order optimization method (gradient descent) finds a global optimizer with perfect fit (zero-loss) in many…
The Deep Operator Network (DeepONet) is a powerful neural operator architecture that uses two neural networks to map between infinite-dimensional function spaces. This architecture allows for the evaluation of the solution field at any…
Modern power systems require fast and accurate dynamic simulations for stability assessment, digital twins, and real-time control, but classical ODE solvers are often too slow for large-scale or online applications. We propose a…
Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision…