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Two combined methods for computing solutions of time-varying semilinear differential-algebraic equations (descriptor systems) are obtained. When constructing the methods, time-varying spectral projectors which can be found numerically are…
Scientific Machine Learning (ML) is gaining momentum as a cost-effective alternative to physics-based numerical solvers in many engineering applications. In fact, scientific ML is currently being used to build accurate and efficient…
Rational solutions of partial differential equations (PDEs) are notoriously difficult to approximate via spectral Fourier methods due to their algebraically slow decay rate. In this work we discuss approximating rational PDE solutions in a…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
Data-driven acceleration of scientific computing workflows has been a high-profile aim of machine learning (ML) for science, with numerical simulation of transient partial differential equations (PDEs) being one of the main applications.…
Particulate Stokesian flows describe the hydrodynamics of rigid or deformable particles in Stokes flows. Due to highly nonlinear fluid-structure interaction dynamics, moving interfaces, and multiple scales, numerical simulations of such…
This work addresses a central challenge in the numerical analysis of the cutoff spatially homogeneous Boltzmann equation: the development of rigorously justified, accurate numerical schemes. We present (i) a novel Fourier spectral method…
Solving Partial Differential Equations (PDEs) is ubiquitous in science and engineering. Computational complexity and difficulty in writing numerical solvers has motivated the development of data-driven machine learning techniques to…
Accurately, efficiently, and stably computing complex fluid flows and their evolution near solid boundaries over long horizons remains challenging. Conventional numerical solvers require fine grids and small time steps to resolve near-wall…
We propose a framework for training neural networks that are coupled with partial differential equations (PDEs) in a parallel computing environment. Unlike most distributed computing frameworks for deep neural networks, our focus is to…
PDEs are central to scientific and engineering modeling, yet designing accurate numerical solvers typically requires substantial mathematical expertise and manual tuning. Recent neural network-based approaches improve flexibility but often…
We propose a novel type of nonlinear solver acceleration for systems of nonlinear partial differential equations (PDEs) that is based on online/adaptive learning. It is applied in the context of multiphase flow in porous media. The proposed…
Computational cardiovascular flow modeling plays a crucial role in understanding blood flow dynamics. While 3D models provide acute details, they are computationally expensive, especially with fluid-structure interaction (FSI) simulations.…
Spectral methods employing non-standard polynomial bases, such as M\"untz polynomials, have proven effective for accurately solving problems with solutions exhibiting low regularity, notably including sub-diffusion equations. However, due…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
Many claims of computational advantages have been made for quantum computing over classical, but they have not been demonstrated for practical problems. Here, we present algorithms for solving time-dependent PDEs, with particular reference…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…