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Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying…

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…

We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) for many-query problems. This work is inspired by the concept of implicit neural representation (INR),…

We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE…

Reduced order modeling lowers the computational cost of solving PDEs by learning a low-order spatial representation from data and dynamically evolving these representations using manifold projections of the governing equations. While…

Physics-based models often involve large systems of parametrized partial differential equations, where design parameters control various properties. However, high-fidelity simulations of such systems on large domains or with high grid…

Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows…

Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state…

The use of reduced-order models (ROMs) in physics-based modeling and simulation almost always involves the use of linear reduced basis (RB) methods such as the proper orthogonal decomposition (POD). For some nonlinear problems, linear RB…

Neural networks have shown promising potential in accelerating the numerical simulation of systems governed by partial differential equations (PDEs). Different from many existing neural network surrogates operating on high-dimensional…

Solving and optimising Partial Differential Equations (PDEs) in geometrically parameterised domains often requires iterative methods, leading to high computational and time complexities. One potential solution is to learn a direct mapping…

Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient…

We develop an unsupervised machine learning algorithm for the automated discovery and identification of traveling waves in spatio-temporal systems governed by partial differential equations (PDEs). Our method uses sparse regression and…

We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical…

Physics-informed neural networks (PINNs) have shown to be an effective tool for solving forward and inverse problems of partial differential equations (PDEs). PINNs embed the PDEs into the loss of the neural network, and this PDE loss is…

Time-dependent partial differential equations are ubiquitous in physics-based modeling, but they remain computationally intensive in many-query scenarios, such as real-time forecasting, optimal control, and uncertainty quantification.…

Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the…

Inverse problems are pervasive mathematical methods in inferring knowledge from observational and experimental data by leveraging simulations and models. Unlike direct inference methods, inverse problem approaches typically require many…

Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…

Randomized neural network (RaNN) methods have been proposed for solving various partial differential equations (PDEs), demonstrating high accuracy and efficiency. However, initializing the fixed parameters remains challenging. Additionally,…