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Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…

Partial differential equations (PDEs) govern a wide variety of dynamical processes in science and engineering, yet obtaining their numerical solutions often requires high-resolution discretizations and repeated evaluations of complex…

This thesis develops numerical and theoretical approaches for understanding and analyzing singularity formation in Partial Differential Equations (PDEs). The singularity formation in the Navier-Stokes Equation (NSE) is famously challenging…

This work proposes an autoencoder neural network as a non-linear generalization of projection-based methods for solving Partial Differential Equations (PDEs). The proposed deep learning architecture presented is capable of generating the…

Programmable optical neural networks (ONNs) can offer high-throughput and energy-efficient solutions for accelerating artificial intelligence (AI) computing. However, existing ONN architectures, typically based on cascaded unitary…

Partial differential equations (PDEs) are often computationally challenging to solve, and in many settings many related PDEs must be be solved either at every timestep or for a variety of candidate boundary conditions, parameters, or…

Predicting outcomes and planning interactions with the physical world are long-standing goals for machine learning. A variety of such tasks involves continuous physical systems, which can be described by partial differential equations…

Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a…

We propose the Factorized Fourier Neural Operator (F-FNO), a learning-based approach for simulating partial differential equations (PDEs). Starting from a recently proposed Fourier representation of flow fields, the F-FNO bridges the…

Meshless methods are commonly used to determine numerical solutions to partial differential equations (PDEs) for problems involving free surfaces and/or complex geometries, approximating spatial derivatives at collocation points via local…

We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in…

This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…

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…

Although deep models have been widely explored in solving partial differential equations (PDEs), previous works are primarily limited to data only with up to tens of thousands of mesh points, far from the million-point scale required by…

Open Computing Language (OpenCL) is a parallel processing language that is ideally suited for running parallel algorithms on Graphical Processing Units (GPUs). In the present work we report on the development of a generic parallel…

Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental…

High-precision scientific simulation faces a long-standing trade-off between computational efficiency and physical fidelity. To address this challenge, we propose NeuralOGCM, an ocean modeling framework that fuses differentiable programming…

In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…

We address a three-tier data-driven approach to solve the inverse problem in complex systems modelling from spatio-temporal data produced by microscopic simulators using machine learning. In the first step, we exploit manifold learning and…

Differential equations (DE) constrained optimization plays a critical role in numerous scientific and engineering fields, including energy systems, aerospace engineering, ecology, and finance, where optimal configurations or control…