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Singularly perturbed partial differential equations arise in many applications, including magnetohydrodynamic duct flows, chemical reaction transport systems, and Poisson Boltzmann electrostatics. These problems are characterized by sharp…

The least squares method with deep neural networks as function parametrization has been applied to solve certain high-dimensional partial differential equations (PDEs) successfully; however, its convergence is slow and might not be…

In the recent years, deep learning approaches have shown much promise in modeling complex systems in the physical sciences. A major challenge in deep learning of PDEs is enforcing physical constraints and boundary conditions. In this work,…

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),…

Sparse regression on a library of candidate features has developed as the prime method to discover the partial differential equation underlying a spatio-temporal data-set. These features consist of higher order derivatives, limiting model…

Discovering governing equations from data is critical for diverse scientific disciplines as they can provide insights into the underlying phenomenon of dynamic systems. This work presents a new representation for governing equations by…

Numerical discretisations of partial differential equations (PDEs) can be written as discrete convolutions, which, themselves, are a key tool in AI libraries and used in convolutional neural networks (CNNs). We therefore propose to…

This work proposes a solution for the problem of training physics-informed networks under partial integro-differential equations. These equations require an infinite or a large number of neural evaluations to construct a single residual for…

We consider the data-driven discovery of governing equations from time-series data in the limit of high noise. The algorithms developed describe an extensive toolkit of methods for circumventing the deleterious effects of noise in the…

Physics-informed neural networks (PINNs) have recently become a powerful tool for solving partial differential equations (PDEs). However, finding a set of neural network parameters that lead to fulfilling a PDE can be challenging and…

In this work, we present a machine learning approach for reducing the error when numerically solving time-dependent partial differential equations (PDE). We use a fully convolutional LSTM network to exploit the spatiotemporal dynamics of…

Uncertainty estimation for unlabeled data is crucial to active learning. With a deep neural network employed as the backbone model, the data selection process is highly challenging due to the potential over-confidence of the model…

Partial differential equation (PDE) models and their associated variational energy formulations are often rotationally invariant by design. This ensures that a rotation of the input results in a corresponding rotation of the output, which…

Multi-query applications such as parameter estimation, uncertainty quantification and design optimization for parameterized PDE systems are expensive due to the high computational cost of high-fidelity simulations. Reduced/Latent state…

Advanced deep learning-based approaches have been actively applied to forecast the spatiotemporal physical dynamics governed by partial differential equations (PDEs), which acts as a critical procedure in tackling many science and…

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for…

On the forefront of scientific computing, Deep Learning (DL), i.e., machine learning with Deep Neural Networks (DNNs), has emerged a powerful new tool for solving Partial Differential Equations (PDEs). It has been observed that DNNs are…

Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by…

Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of…

In recent years, deep learning for modeling physical phenomena which can be described by partial differential equations (PDEs) have received significant attention. For example, for learning Hamiltonian mechanics, methods based on deep…