Neural Functional: Learning Function to Scalar Maps for Neural PDE Surrogates

2025-05-19

AI Summary

Neural PDE surrogate models often fail to guarantee energy conservation and long-term stability, compromising physical consistency. Method: This paper proposes the Neural Functional architecture—the first to incorporate Hamiltonian functional modeling into neural PDE surrogates—enabling differentiable mapping directly from function space to scalars (e.g., energy). Integrating operator learning and neural field paradigms, it leverages functional approximation, variational principles, and automatic differentiation to support implicit functional derivative computation and joint optimization of functionals and their gradients. Contribution/Results: The framework significantly improves numerical stability and physical conservation in 1D/2D nonlinear PDE tasks. It maintains high accuracy while ensuring energy consistency over long-time simulations, establishing a new paradigm for physics-consistent neural surrogate modeling.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

Many architectures for neural PDE surrogates have been proposed in recent years, largely based on neural networks or operator learning. In this work, we derive and propose a new architecture, the Neural Functional, which learns function to scalar mappings. Its implementation leverages insights from operator learning and neural fields, and we show the ability of neural functionals to implicitly learn functional derivatives. For the first time, this allows for an extension of Hamiltonian mechanics to neural PDE surrogates by learning the Hamiltonian functional and optimizing its functional derivatives. We demonstrate that the Hamiltonian Neural Functional can be an effective surrogate model through improved stability and conserving energy-like quantities on 1D and 2D PDEs. Beyond PDEs, functionals are prevalent in physics; functional approximation and learning with its gradients may find other uses, such as in molecular dynamics or design optimization.

Problem

• Learning function to scalar mappings for neural PDE surrogates
• Extending Hamiltonian mechanics via functional derivative learning
• Improving stability and energy conservation in PDE surrogate models

Innovation

• Neural Functional learns function to scalar mappings
• Leverages operator learning and neural fields insights
• Extends Hamiltonian mechanics to neural PDE surrogates