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

Haoning Dang, Shi Jin, Fei Wang

TL;DR

An Adaptive-Growth Randomized Neural Network method for computing multivalued solutions of nonlinear first-order PDEs with hyperbolic characteristics, including quasilinear hyperbolic balance laws and Hamilton--Jacobi equations is proposed.

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.

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

TL;DR

An Adaptive-Growth Randomized Neural Network method for computing multivalued solutions of nonlinear first-order PDEs with hyperbolic characteristics, including quasilinear hyperbolic balance laws and Hamilton--Jacobi equations is proposed.

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.
Paper Structure (20 sections, 4 theorems, 86 equations, 17 figures, 5 tables)

This paper contains 20 sections, 4 theorems, 86 equations, 17 figures, 5 tables.

Table of Contents

  1. Introduction
  2. First-Order PDEs and Level-Set Method
  3. Hamilton--Jacobi Equations: Level-Set Formulation for Gradient Fields
  4. Scalar Hyperbolic Balance Laws: Level-Set Formulation for the Solution
  5. Adaptive-Growth Randomized Neural Networks
  6. AG-RaNN with the Layer Growth Strategy
  7. Adaptive Collocation Points
  8. Error-Indicator Points
  9. Numerical Analysis
  10. Graph Norm Equivalence
  11. Optimization Term
  12. Layer Growth
  13. Numerical Experiments
  14. Zero Level Set
  15. Scalar-valued case
  16. ...and 5 more sections

Key Result

Theorem 4.3

Under Assumption ass:char, consider the operator $\mathcal{G}:V_2\to Y$ given by eq:ls-operator and the boundary operator $\mathcal{B}:V_2\to Z$ defined in eq:ls-bc. Then there exist constants $C_L,C_U>0$, depending only on $D$, $C_a$, $c_0$, and $C_0$, such that for all $\phi\in H^1(D)$

Figures (17)

  • Figure 1: Schematic diagram of adaptive collocation points.
  • Figure 2: The iterative method to find the zero level set.
  • Figure 3: AG-RaNN method results for Case 1 of Example \ref{['ex1']}. Columns 1-3 correspond to $t=0,0.5,1$, and column 4 shows the zero level set. Rows (top to bottom) use normal collocation points, collocation set $\Lambda_A$ ($N_A^I=39784$, $N_A^B=836$), and the layer growth strategy, respectively. Running times: $(t_1,t_2,t_3)=(2.45,1.63,7.23)$.
  • Figure 4: AG-RaNN method results for Case 2 of Example \ref{['ex1']}. Columns 1-3 correspond to $t=0,0.5,1$, and column 4 shows the zero level set. Rows (top to bottom) use normal collocation points, collocation set $\Lambda_A$ ($N_A^I=33051$, $N_A^B=799$), and the layer growth strategy, respectively. Running times: $(t_1,t_2,t_3)=(2.47,1.38,8.70)$.
  • Figure 5: AG-RaNN method results for Case 3 of Example \ref{['ex1']}. Columns 1-3 correspond to $t=0,0.5,1$, and column 4 shows the zero level set. Rows (top to bottom) use normal collocation points, collocation set $\Lambda_A$ ($N_A^I=96122$, $N_A^B=1685$), and the layer growth strategy, respectively. Running times: $(t_1,t_2,t_3)=(2.47,5.52,23.28)$.
  • ...and 12 more figures

Theorems & Definitions (13)

  • Remark 3.1
  • Remark 3.2
  • Remark 4.2
  • Theorem 4.3: Graph norm equivalence for the level-set equations
  • Remark 4.5
  • Theorem 4.6: Approximation Error
  • Theorem 4.7: Statistical Component Bound
  • Theorem 4.9: Theorem 5.3.1 in Golub2013matrix
  • Example 1: 1D inviscid forced Burgers' Equation
  • Example 2: 2D Burgers' Equation
  • ...and 3 more