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Neural operators aim to approximate the solution operator of a system of differential equations purely from data. They have shown immense success in modeling complex dynamical systems across various domains. However, the occurrence of…

Many large scale problems in computational fluid dynamics such as uncertainty quantification, Bayesian inversion, data assimilation and PDE constrained optimization are considered very challenging computationally as they require a large…

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…

Operator learning aims to discover properties of an underlying dynamical system or partial differential equation (PDE) from data. Here, we present a step-by-step guide to operator learning. We explain the types of problems and PDEs amenable…

The use of neural networks to approximate partial differential equations (PDEs) has gained significant attention in recent years. However, the approximation of PDEs with localised phenomena, e.g., sharp gradients and singularities, remains…

Industrial design evaluation often relies on high-fidelity simulations of governing partial differential equations (PDEs). While accurate, these simulations are computationally expensive, making dense exploration of design spaces…

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…

Modeling the traffic dynamics is essential for understanding and predicting the traffic spatiotemporal evolution. However, deriving the partial differential equation (PDE) models that capture these dynamics is challenging due to their…

Residual neural networks can be viewed as the forward Euler discretization of an Ordinary Differential Equation (ODE) with a unit time step. This has recently motivated researchers to explore other discretization approaches and train ODE…

Neuroscientists fit morphologically and biophysically detailed neuron simulations to physiological data, often using evolutionary algorithms. However, such gradient-free approaches are computationally expensive, making convergence slow when…

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…

Neural Ordinary Differential Equations (NODEs) are a novel neural architecture, built around initial value problems with learned dynamics which are solved during inference. Thought to be inherently more robust against adversarial…

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…

Neural operator models for solving partial differential equations (PDEs) often rely on global mixing mechanisms-such as spectral convolutions or attention-which tend to oversmooth sharp local dynamics and introduce high computational cost.…

Learning PDE dynamics from limited data with unknown physics is challenging. Existing neural PDE solvers either require large datasets or rely on known physics (e.g., PDE residuals or handcrafted stencils), leading to limited applicability.…

Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential…

Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation…

Neural ordinary differential equations (neural ODEs) have emerged as a novel network architecture that bridges dynamical systems and deep learning. However, the gradient obtained with the continuous adjoint method in the vanilla neural ODE…

In scientific and engineering applications, solving partial differential equations (PDEs) across various parameters and domains normally relies on resource-intensive numerical methods. Neural operators based on deep learning offered a…

Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world. At least two challenges exist for PINNs at…