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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…

The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary…

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…

We present a neural network-based method for learning scalar hyperbolic conservation laws. Our method replaces the traditional numerical flux in finite volume schemes with a trainable neural network while preserving the conservative…

Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…

Physics-Informed Neural Networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs). PINNs are based on simple architectures, and learn the behavior of complex…

Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods…

Optimal control schemes have achieved remarkable performance in numerous engineering applications. However, they typically require high computational cost, which has limited their use in real-world engineering systems with fast dynamics…

Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their…

Neural PDE solvers offer a powerful tool for modeling complex dynamical systems, but often struggle with error accumulation over long time horizons and maintaining stability and physical consistency. We introduce a multiscale implicit…

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…

Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning.…

Neural operators effectively solve PDE problems from data without knowing the explicit equations, which learn the map from the input sequences of observed samples to the predicted values. Most existing works build the model in the original…

This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel…

Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks…

Learning time-dependent partial differential equations (PDEs) that govern evolutionary observations is one of the core challenges for data-driven inference in many fields. In this work, we propose to capture the essential dynamics of…

Deep neural operators are recognized as an effective tool for learning solution operators of complex partial differential equations (PDEs). As compared to laborious analytical and computational tools, a single neural operator can predict…

This paper develops an extension of infinite-dimensional backstepping method for parabolic and hyperbolic systems in one spatial dimension with two actuators. Typically, PDE backstepping is applied in 1-D domains with an actuator at one…

We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an…

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…