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

• URL: http://arxiv.org/abs/2603.01093v1
• Date: Sun, 01 Mar 2026 13:16:25 GMT
• Title: Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics
• Authors: Haoning Dang, Shi Jin, Fei Wang,
• Abstract summary: This paper proposes an Adaptive-Growth Randomized Neural Network (AG-RaNN) method for computing multivalued solutions of nonlinear first-order PDEs with hyperbolic characteristics.<n>Such solutions arise in geometric optics, seismic waves, semiclassical limit of quantum dynamics and high frequency limit of linear waves.
• Score: 38.23142730599331
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations [14.314318817152165]
  We develop a novel framework to solve the incompressible magnetohydrodynamic (MHD) equations.<n>It preserves the strong nonlinearity and dual divergence-free constraints.<n>It achieves higher accuracy, faster convergence, and exact enforcement of divergence-free constraints.
  arXiv  Detail & Related papers  (2026-03-01T13:42:28Z)
• UltraLIF: Fully Differentiable Spiking Neural Networks via Ultradiscretization and Max-Plus Algebra [0.0]
  Spiking Neural Networks (SNNs) offer energy-efficient, biologically plausible computation but suffer from non-differentiable spike generation.<n>This paper introduces UltraLIF, a principled framework that replaces surrogate gradients with ultradiscretization.<n>Experiments on six benchmarks spanning static images, neuromorphic vision, and audio demonstrate improvements over surrogate gradient baselines.
  arXiv  Detail & Related papers  (2026-02-10T18:21:54Z)
• Neural Optimal Transport Meets Multivariate Conformal Prediction [58.43397908730771]
  We propose a framework for conditional vectorile regression (CVQR)<n>CVQR combines neural optimal transport with quantized optimization, and apply it to predictions.
  arXiv  Detail & Related papers  (2025-09-29T19:50:19Z)
• Physics-informed neural networks for high-dimensional solutions and snaking bifurcations in nonlinear lattices [0.0]
  This paper introduces a framework based on physics-informed neural networks (PINNs) for addressing key challenges in nonlinear lattices.<n>We first employ PINNs to approximate solutions of nonlinear systems arising from lattice models, using the Levenberg-Marquardt algorithm.<n>We then extend the method by coupling PINNs with a continuation approach to compute snaking bifurcation diagrams.<n>For linear stability analysis, we adapt PINNs to compute eigenvectors, introducing output constraints to enforce positivity, in line with Sturm-Liouville theory.
  arXiv  Detail & Related papers  (2025-07-13T20:41:55Z)
• S-Crescendo: A Nested Transformer Weaving Framework for Scalable Nonlinear System in S-Domain Representation [4.945568106952893]
  S-Crescendo is a nested transformer weaving framework that synergizes S-domain with neural operators for scalable time-domain prediction.<n>Our method achieves up to 0.99 test-set ($R2$) accuracy against HSPICE golden waveforms and simulation accelerates by up to 18(X)
  arXiv  Detail & Related papers  (2025-05-17T05:06:58Z)
• The Convex Landscape of Neural Networks: Characterizing Global Optima and Stationary Points via Lasso Models [75.33431791218302]
  Deep Neural Network Network (DNN) models are used for programming purposes. In this paper we examine the use of convex neural recovery models. We show that all the stationary non-dimensional objective objective can be characterized as the standard a global subsampled convex solvers program. We also show that all the stationary non-dimensional objective objective can be characterized as the standard a global subsampled convex solvers program.
  arXiv  Detail & Related papers  (2023-12-19T23:04:56Z)
• A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors [60.489902135153415]
  This paper introduces a variational iterative alternating scheme for hierarchical inverse problems with gamma hyperpriors. The proposed variational inference approach yields accurate reconstruction, provides meaningful uncertainty quantification, and is easy to implement.
  arXiv  Detail & Related papers  (2021-11-26T06:33:29Z)
• Physics and Equality Constrained Artificial Neural Networks: Application to Partial Differential Equations [1.370633147306388]
  Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE) Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach. We propose a versatile framework that can tackle both inverse and forward problems.
  arXiv  Detail & Related papers  (2021-09-30T05:55:35Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.