The recent developments in the field of solving partial differential equations (PDEs) using deep learning techniques have shown a significant shift towards addressing the challenges of high-dimensionality, spectral biases, and the integration of physical laws into neural network models. Innovations in this area focus on enhancing the accuracy, efficiency, and applicability of Physics-Informed Neural Networks (PINNs) and related methodologies. A notable trend is the development of novel frameworks that combine data-driven approaches with physical constraints to solve complex PDEs, including those arising in quantum dynamics, optimal transport problems, and subsurface flow modeling. These advancements are characterized by the introduction of new techniques such as convex neural networks for enforcing solution properties, quantum algorithms for highly-oscillatory transport equations, and hybrid models that integrate multiscale basis functions with deep learning for accurate reconstruction of physical phenomena. Additionally, there is a growing emphasis on automating the process of PDE surrogation and improving the computational efficiency of neural network-based solutions through innovative sampling methods and regularization techniques. The field is also witnessing a surge in the application of randomized neural networks and physics-aware signal recovery methods, which promise to alleviate the curse of dimensionality and enhance the fidelity of solutions to PDEs governed by complex physical laws.

Noteworthy Papers

• Convex Physics Informed Neural Networks for the Monge-Ampère Optimal Transport Problem: Introduces convex neural networks to enforce solution convexity in solving the Monge-Ampère equation, demonstrating effectiveness in optimal transport scenarios.
• Quantum simulation of a class of highly-oscillatory transport equations via Schrödingerisation: Presents a quantum algorithm for highly-oscillatory transport equations, ensuring uniform error estimates independent of wavelength.
• Deep Operator Networks for Bayesian Parameter Estimation in PDEs: Combines DeepONets with PINNs for robust PDE solutions and parameter estimation, incorporating Bayesian training for uncertainty quantification.
• An Imbalanced Learning-based Sampling Method for Physics-informed Neural Networks: Introduces RSmote, a sampling technique that improves PINN performance by targeting high-residual regions and reducing memory usage.
• PINNsAgent: Automated PDE Surrogation with Large Language Models: Leverages LLMs to automate the surrogation process, enhancing the efficiency and accuracy of PINNs-based solutions.
• Efficient PINNs: Multi-Head Unimodular Regularization of the Solutions Space: Proposes a multihead training approach with unimodular regularization to improve PINN efficiency in solving nonlinear, coupled, and multiscale differential equations.
• Hybrid Two-Stage Reconstruction of Multiscale Subsurface Flow with Physics-informed Residual Connected Neural Operator: Develops a hybrid framework for accurate reconstruction of subsurface flow, integrating multiscale basis functions with deep learning.
• Physics-Aware Sparse Signal Recovery Through PDE-Governed Measurement Systems: Introduces PA-ISTA, a physics-aware iterative shrinkage-thresholding algorithm for improved signal recovery in PDE-governed systems.