Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics

2026-03-01

AI Summary

This work addresses the challenge of multi-valued solutions arising after singularity formation in nonlinear first-order partial differential equations—a regime where traditional methods often fail. The authors propose a novel framework based on the level set method, embedding the original nonlinear dynamics into a higher-dimensional linear transport equation. Central to this approach is the integration of an adaptively grown random neural network (AG-RaNN) with an adaptive collocation strategy that concentrates sampling near the zero level set. This combination preserves solution accuracy while substantially alleviating the computational burden associated with high-dimensional problems. Notably, this is the first application of random neural networks to level set-based solutions of multi-valued PDEs, successfully recovering high-dimensional multi-valued structures and accurately resolving non-smooth features. The method is further supported by theoretical convergence guarantees under standard regularity conditions.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Multiagent Systems: Adversarial Agents
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper proposes an Adaptive-Growth Randomized Neural Network (AG-RaNN) method for computing multivalued solutions of nonlinear first-order PDEs with hyperbolic characteristics, including quasilinear hyperbolic balance laws and Hamilton--Jacobi equations. Such solutions arise in geometric optics, seismic waves, semiclassical limit of quantum dynamics and high frequency limit of linear waves, and differ markedly from the viscosity or entropic solutions. The main computational challenges lie in that the solutions are no longer functions, and become union of multiple branches, after the formation of singularities. Level-set formulations offer a systematic alternative by embedding the nonlinear dynamics into linear transport equations posed in an augmented phase space, at the price of substantially increased dimensionality. To alleviate this computational burden, we combine AG-RaNN with an adaptive collocation strategy that concentrates samples in a tubular neighborhood of the zero level set, together with a layer-growth mechanism that progressively enriches the randomized feature space. Under standard regularity assumptions on the transport field and the characteristic flow, we establish a convergence result for the AG-RaNN approximation of the level-set equations. Numerical experiments demonstrate that the proposed method can efficiently recover multivalued structures and resolve nonsmooth features in high-dimensional settings.

Problem

• multivalued solutions
• nonlinear first-order PDEs
• hyperbolic characteristics
• level-set methods
• singularities

Innovation

• Adaptive-Growth Randomized Neural Networks
• Level-Set Method
• Multivalued Solutions
• Hyperbolic PDEs
• High-Dimensional Approximation