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Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for…

Learning dynamics governed by differential equations is crucial for predicting and controlling the systems in science and engineering. Neural Ordinary Differential Equation (NODE), a deep learning model integrated with differential…

Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…

Turbulence simulation with classical numerical solvers requires high-resolution grids to accurately resolve dynamics. Here we train learned simulators at low spatial and temporal resolutions to capture turbulent dynamics generated at high…

Since the seminal work of [9] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for…

Pre-training has been investigated to improve the efficiency and performance of training neural operators in data-scarce settings. However, it is largely in its infancy due to the inherent complexity and diversity, such as long…

The data-driven discovery of partial differential equations (PDEs) consistent with spatiotemporal data is experiencing a rebirth in machine learning research. Training deep neural networks to learn such data-driven partial differential…

The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…

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…

Accurately solving partial differential equations (PDEs) is critical to understanding complex scientific and engineering phenomena, yet traditional numerical solvers are computationally expensive. Surrogate models offer a more efficient…

Reliable physics simulation demands two capabilities that today's neural PDE solvers do not deliver together: generalization across heterogeneous PDE families, and stability under long autoregressive rollouts. Deterministic operators…

Neural networks have been identified as powerful tools for the study of complex systems. A noteworthy example is the neural network differential equation (NN DE) solver, which can provide functional approximations to the solutions of a wide…

Artificial intelligence (AI) for fluid mechanics has become attractive topic. High-fidelity data is one of most critical issues for the successful applications of AI in fluid mechanics, however, it is expensively obtained or even…

Generative models have become a powerful tool for synthesizing training data in computer vision tasks. Current approaches solely focus on aligning generated images with the target dataset distribution. As a result, they capture only the…

In this era of diverse and heterogeneous computer architectures, the programmability issues, such as productivity and portable efficiency, are crucial to software development and algorithm design. One way to approach the problem is to step…

The use of quantum computing for machine learning is among the most exciting prospective applications of quantum technologies. However, machine learning tasks where data is provided can be considerably different than commonly studied…

The limited availability of high-quality training data poses a major obstacle in data-driven PDE solving, where expensive data collection and resolution constraints severely impact the ability of neural operator networks to learn and…

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…

Efficiently solving Poisson equations on complex, irregular domains remains a fundamental challenge in scientific computing, as classical iterative solvers often suffer from prohibitive runtime due to ill-conditioned systems. While neural…

Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…