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Surrogate models that combine dimensionality reduction and regression techniques are essential to reduce the need for costly high-fidelity computational fluid dynamics data. New approaches using $\beta$-Variational Autoencoder ($\beta$-VAE)…

Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…

Partial differential equation (PDE)-constrained optimization arises in many scientific and engineering domains, such as energy systems, fluid dynamics and material design. In these problems, the decision variables (e.g., control inputs or…

Deep neural networks (DNNs), especially physics-informed neural networks (PINNs), have recently become a new popular method for solving forward and inverse problems governed by partial differential equations (PDEs). However, these methods…

Neural operators (NOs) provide a new paradigm for efficiently solving partial differential equations (PDEs), but their training depends on costly high-fidelity data from numerical solvers, limiting applications in complex systems. We…

Classical methods for model order selection often fail in scenarios with low SNR or few snapshots. Deep learning-based methods are promising alternatives for such challenging situations as they compensate lack of information in the…

A Theory-guided Auto-Encoder (TgAE) framework is proposed for surrogate construction and is further used for uncertainty quantification and inverse modeling tasks. The framework is built based on the Auto-Encoder (or Encoder-Decoder)…

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…

Solving partial differential equations (PDEs) is a fundamental problem in science and engineering. While neural PDE solvers can be more efficient than established numerical solvers, they often require large amounts of training data that is…

Learning the full family of solutions to parameterized partial differential equations (PDEs) is a central challenge to our ability to model the behavior of heterogeneous systems, with a variety of fundamental and application-oriented…

Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for solving partial differential equations (PDEs). However, training PINNs from scratch is often computationally intensive and time-consuming. To address this problem,…

Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems.…

Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data…

Fast and accurate surrogates for physics-driven partial differential equations (PDEs) are essential in fields such as aerodynamics, porous media design, and flow control. However, many transformer-based models and existing neural operators…

In this paper, an innovative Physical Model-driven Neural Network (PMNN) method is proposed to solve time-fractional differential equations. It establishes a temporal iteration scheme based on physical model-driven neural networks which…

Physics-informed neural networks (PINNs) have recently emerged as a promising way to compute the solutions of partial differential equations (PDEs) using deep neural networks. However, despite their significant success in various fields, it…

Adjoint-based optimization methods are attractive for aerodynamic shape design primarily due to their computational costs being independent of the dimensionality of the input space and their ability to generate high-fidelity gradients that…

Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and…

We propose a method combining boundary integral equations and neural networks (BINet) to solve partial differential equations (PDEs) in both bounded and unbounded domains. Unlike existing solutions that directly operate over original PDEs,…

Several forms for constructing novel physics-informed neural-networks (PINN) for the solution of partial-differential-algebraic equations based on derivative operator splitting are proposed, using the nonlinear Kirchhoff rod as a prototype…