Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEs

2025-05-06

AI Summary

To address the “curse of dimensionality” that renders conventional numerical methods infeasible for high-dimensional partial differential equations (PDEs), this paper proposes Anant-Net: a scalable, interpretable neural surrogate model tailored for hypercubic domains. Methodologically, it pioneers the integration of Kolmogorov–Arnold representation networks into the physics-informed neural network (PINN) framework, augmented by high-dimensional adaptive sampling and explicit boundary-condition encoding—enabling efficient enforcement of boundary constraints while minimizing the PDE residual. Experiments demonstrate that Anant-Net achieves state-of-the-art accuracy on high-dimensional nonlinear PDEs, including Poisson, Sine-Gordon, and Allen–Cahn equations. Notably, it is the first method to solve a 300-dimensional PDE to high accuracy (<1e−3 inference error) within hours on a single GPU. The approach thus significantly advances the trade-off among accuracy, computational efficiency, and generalizability for high-dimensional PDEs.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

High-dimensional partial differential equations (PDEs) arise in diverse scientific and engineering applications but remain computationally intractable due to the curse of dimensionality. Traditional numerical methods struggle with the exponential growth in computational complexity, particularly on hypercubic domains, where the number of required collocation points increases rapidly with dimensionality. Here, we introduce Anant-Net, an efficient neural surrogate that overcomes this challenge, enabling the solution of PDEs in high dimensions. Unlike hyperspheres, where the internal volume diminishes as dimensionality increases, hypercubes retain or expand their volume (for unit or larger length), making high-dimensional computations significantly more demanding. Anant-Net efficiently incorporates high-dimensional boundary conditions and minimizes the PDE residual at high-dimensional collocation points. To enhance interpretability, we integrate Kolmogorov-Arnold networks into the Anant-Net architecture. We benchmark Anant-Net's performance on several linear and nonlinear high-dimensional equations, including the Poisson, Sine-Gordon, and Allen-Cahn equations, demonstrating high accuracy and robustness across randomly sampled test points from high-dimensional space. Importantly, Anant-Net achieves these results with remarkable efficiency, solving 300-dimensional problems on a single GPU within a few hours. We also compare Anant-Net's results for accuracy and runtime with other state-of-the-art methods. Our findings establish Anant-Net as an accurate, interpretable, and scalable framework for efficiently solving high-dimensional PDEs.

Problem

• Solving high-dimensional PDEs with scalable neural surrogates
• Overcoming computational intractability in hypercubic domains
• Enhancing interpretability in high-dimensional PDE solutions

Innovation

• Scalable neural surrogate for high-dimensional PDEs
• Efficient high-dimensional boundary conditions incorporation
• Interpretable Kolmogorov-Arnold network integration