The field of solving partial differential equations (PDEs) is witnessing significant advancements with the integration of neural operators. Recent developments focus on enhancing the accuracy, efficiency, and robustness of neural operator-based methods for solving PDEs. A key direction is the incorporation of physical laws and constraints into the neural network architecture, ensuring that the solutions respect the underlying physics of the problem. This approach has led to improved performance in solving various types of PDEs, including parametric and nonlinear equations. Another important trend is the development of hybrid methods that combine the strengths of classical numerical solvers with the flexibility of neural networks. These hybrid approaches aim to leverage the advantages of both worlds, providing reliable and accurate solutions while reducing computational costs. Noteworthy papers in this area include the proposal of a Physics-Informed Time-Integrated DeepONet, which achieves high-accuracy inference for time-dependent PDEs, and the introduction of a Global-Focal Neural Operator for solving PDEs on arbitrary geometries, which enforces simultaneous global and local feature learning and fusion.
Advancements in Solving Partial Differential Equations with Neural Operators
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Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method
Accelerating Conjugate Gradient Solvers for Homogenization Problems with Unitary Neural Operators
DD-DeepONet: Domain decomposition and DeepONet for solving partial differential equations in three application scenarios
Physics-Embedded Neural ODEs for Sim2Real Edge Digital Twins of Hybrid Power Electronics Systems
Bridging ocean wave physics and deep learning: Physics-informed neural operators for nonlinear wavefield reconstruction in real-time
Reduced Order Data-driven Twin Models for Nonlinear PDEs by Randomized Koopman Orthogonal Decomposition and Explainable Deep Learning
VAE-DNN: Energy-Efficient Trainable-by-Parts Surrogate Model For Parametric Partial Differential Equations
Convolutional autoencoders for the reconstruction of three-dimensional interfacial multiphase flows
Extreme Event Precursor Prediction in Turbulent Dynamical Systems via CNN-Augmented Recurrence Analysis
GFocal: A Global-Focal Neural Operator for Solving PDEs on Arbitrary Geometries
Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference