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1. Solving spatial-temporal PDEs with arbitrary boundary conditions using physics-constrained convolutional recurrent neural networks

   https://doi.org/10.1016/j.neucom.2025.129917  

   Li, Guangfa; Lu, Yanglong; Liu, Dehao (June 2025, Neurocomputing)

   The inception of physics-constrained or physics-informed machine learning represents a paradigm shift, addressing the challenges associated with data scarcity and enhancing model interpretability. This innovative approach incorporates the fundamental laws of physics as constraints, guiding the training process of machine learning models. In this work, the physics-constrained convolutional recurrent neural network is further extended for solving spatial-temporal partial differential equations with arbitrary boundary conditions. Two notable advancements are introduced: the implementation of boundary conditions as soft constraints through finite difference-based differentiation, and the establishment of an adaptive weighting mechanism for the optimal allocation of weights to various losses. These enhancements significantly augment the network's ability to manage intricate boundary conditions and expedite the training process. The efficacy of the proposed model is validated through its application to two-dimensional phase transition, fluid dynamics, and reaction-diffusion problems, which are pivotal in materials modeling. Compared to traditional physics-constrained neural networks, the physics-constrained convolutional recurrent neural network demonstrates a tenfold increase in prediction accuracy within a similar computational budget. Moreover, the model's exceptional performance in extrapolating solutions for the Burgers' equation underscores its utility. Therefore, this research establishes the physics-constrained recurrent neural network as a viable surrogate model for sophisticated spatial-temporal PDE systems, particularly beneficial in scenarios plagued by sparse and noisy datasets. 

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2. Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs

   https://doi.org/10.1515/cmam-2021-0118  

   Shi, Tianyi; Antil, Harbir; Kouri, Drew P. (February 2022, Computational Methods in Applied Mathematics)

   Abstract Fractional PDEs have recently found several geophysics and imaging science applications due to their nonlocal nature and their flexibility in capturing sharp transitions across interfaces.However, this nonlocality makes it challenging to design efficient solvers for such problems.In this paper, we introduce a spectral method based on an ultraspherical polynomial discretization of the Caffarelli–Silvestre extension to solve such PDEs on rectangular and disk domains.We solve the discretized problem using tensor equation solvers and thus can solve higher-dimensional PDEs.In addition, we introduce both serial and parallel domain decomposition solvers.We demonstrate the numerical performance of our methods on a 3D fractional elliptic PDE on a cube as well as an application to optimization problems with fractional PDE constraints. 

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3. Scalable Learning for Spatiotemporal Mean Field Games Using Physics-Informed Neural Operator

   https://doi.org/10.3390/math12060803  

   Liu, Shuo; Chen, Xu; Di, Xuan (March 2024, Mathematics)

   This paper proposes a scalable learning framework to solve a system of coupled forward–backward partial differential equations (PDEs) arising from mean field games (MFGs). The MFG system incorporates a forward PDE to model the propagation of population dynamics and a backward PDE for a representative agent’s optimal control. Existing work mainly focus on solving the mean field game equilibrium (MFE) of the MFG system when given fixed boundary conditions, including the initial population state and terminal cost. To obtain MFE efficiently, particularly when the initial population density and terminal cost vary, we utilize a physics-informed neural operator (PINO) to tackle the forward–backward PDEs. A learning algorithm is devised and its performance is evaluated on one application domain, which is the autonomous driving velocity control. Numerical experiments show that our method can obtain the MFE accurately when given different initial distributions of vehicles. The PINO exhibits both memory efficiency and generalization capabilities compared to physics-informed neural networks (PINNs). 

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4. Learning to Accelerate Partial Differential Equations via Latent Global Evolution

   Wu, Tailin; Maruyama, Takashi; Leskovec, Jure (January 2022, Advances in neural information processing systems)

   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. However, both classical solvers and recent deep learning-based surrogate models are typically extremely computationally intensive, because of their local evolution: they need to update the state of each discretized cell at each time step during inference. Here we develop Latent Evolution of PDEs (LE-PDE), a simple, fast and scalable method to accelerate the simulation and inverse optimization of PDEs. LE-PDE learns a compact, global representation of the system and efficiently evolves it fully in the latent space with learned latent evolution models. LE-PDE achieves speedup by having a much smaller latent dimension to update during long rollout as compared to updating in the input space. We introduce new learning objectives to effectively learn such latent dynamics to ensure long-term stability. We further introduce techniques for speeding-up inverse optimization of boundary conditions for PDEs via backpropagation through time in latent space, and an annealing technique to address the non-differentiability and sparse interaction of boundary conditions. We test our method in a 1D benchmark of nonlinear PDEs, 2D Navier-Stokes flows into turbulent phase and an inverse optimization of boundary conditions in 2D Navier-Stokes flow. Compared to state-of-the-art deep learning-based surrogate models and other strong baselines, we demonstrate up to 128x reduction in the dimensions to update, and up to 15x improvement in speed, while achieving competitive accuracy. 

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5. Differential Walk on Spheres

   Miller, Bailey; Sawhney, Rohan; Crane, Keenan; Gkioulekas, Ioannis (December 2024, ACM Transactions on Graphics)

   We introduce a Monte Carlo method for computing derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions). Derivatives can be evaluated at arbitrary points, without performing a global solve or constructing a volumetric grid or mesh. The method is hence well suited to inverse problems with complex geometry, such as PDE-constrained shape optimization. Like other walk on spheres (WoS) algorithms, our method is trivial to parallelize, and is agnostic to boundary representation (meshes, splines, implicit surfaces, etc.), supporting large topological changes. We focus in particular on screened Poisson equations, which model diverse problems from scientific and geometric computing. As in differentiable rendering, we jointly estimate derivatives with respect to all parameters—hence, cost does not grow significantly with parameter count. In practice, even noisy derivative estimates exhibit fast, stable convergence for stochastic gradient-based optimization, as we show through examples from thermal design, shape from diffusion, and computer graphics. 

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