<p>In this work, we propose a framework for solving second-order nonlinear integrable systems in the spirit of the random feature method (RFM), which demonstrates high efficiency and high precision numerical performance. Unlike other RFMs for solving linear partial differential equations (PDEs) or a few relatively classical and simple nonlinear PDEs, our approach focuses on solving second-order nonlinear integrable systems by leveraging partition of unity technology, deep neural network, random feature functions. Once considering second-order nonlinear integrable system with initial-boundary value condition and incorporating appropriate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>κ</mi> </msup> </math></EquationSource> </InlineEquation> continuity conditions, we construct a novel assembly matrix with implicit optimization coefficient and convert the solving problem of second-order nonlinear integrable system into corresponding nonlinear least squares problem. Different from popular physics-informed neural network, it avoids backpropagation-based optimization of loss function and instead solves the equivalent set of nonlinear equations with high precision by means of a nonlinear least squares implementation. The numerical experiments focus on solving several important and representative second-order nonlinear integrable systems, which possess unique mathematical structures, important physical significance and rich localized wave solutions. These include the single solitary wave and two-solitary wave solutions of the Burgers equation, kink and kink-kink solutions of the Sine-Gordon (SG) equation, pulse solution of the short pulse equation, and single-soliton solutions of the coupled Kraenkel-Manna-Merle system. The numerical results demonstrate that the RFM possess a significant advantage in solving localized wave solutions of second-order nonlinear integrable systems with numerical accuracy several orders of magnitude ahead of other deep learning methods, and exhibits comparable or superior performance compared to traditional numerical method. Specifically, for integrable model with easy-to-generate grids (e.g., Burgers equation), RFM shows solving capabilities that rival or even surpass traditional numerical method, while for integrable models with challenging grid generation (e.g., SG equation), RFM outperforms physics-informed neural network. The innovation of this work lies in the first to successfully apply RFM to solve second-order nonlinear integrable systems, offering both theoretical reference and experimental support for studying the numerical localized wave solutions of higher-order and higher-dimensional nonlinear integrable systems, and providing novel approach for solving integrable systems and synthesizing integrable system theory.</p>

Random Feature Method Solving Second-Order Nonlinear Integrable Systems

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发表日期：

2026年7月27日

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Juncai Pu,
Yuhang Wu,
Yong Chen

摘要

In this work, we propose a framework for solving second-order nonlinear integrable systems in the spirit of the random feature method (RFM), which demonstrates high efficiency and high precision numerical performance. Unlike other RFMs for solving linear partial differential equations (PDEs) or a few relatively classical and simple nonlinear PDEs, our approach focuses on solving second-order nonlinear integrable systems by leveraging partition of unity technology, deep neural network, random feature functions. Once considering second-order nonlinear integrable system with initial-boundary value condition and incorporating appropriate $C^\kappa $ $C^{κ}$ continuity conditions, we construct a novel assembly matrix with implicit optimization coefficient and convert the solving problem of second-order nonlinear integrable system into corresponding nonlinear least squares problem. Different from popular physics-informed neural network, it avoids backpropagation-based optimization of loss function and instead solves the equivalent set of nonlinear equations with high precision by means of a nonlinear least squares implementation. The numerical experiments focus on solving several important and representative second-order nonlinear integrable systems, which possess unique mathematical structures, important physical significance and rich localized wave solutions. These include the single solitary wave and two-solitary wave solutions of the Burgers equation, kink and kink-kink solutions of the Sine-Gordon (SG) equation, pulse solution of the short pulse equation, and single-soliton solutions of the coupled Kraenkel-Manna-Merle system. The numerical results demonstrate that the RFM possess a significant advantage in solving localized wave solutions of second-order nonlinear integrable systems with numerical accuracy several orders of magnitude ahead of other deep learning methods, and exhibits comparable or superior performance compared to traditional numerical method. Specifically, for integrable model with easy-to-generate grids (e.g., Burgers equation), RFM shows solving capabilities that rival or even surpass traditional numerical method, while for integrable models with challenging grid generation (e.g., SG equation), RFM outperforms physics-informed neural network. The innovation of this work lies in the first to successfully apply RFM to solve second-order nonlinear integrable systems, offering both theoretical reference and experimental support for studying the numerical localized wave solutions of higher-order and higher-dimensional nonlinear integrable systems, and providing novel approach for solving integrable systems and synthesizing integrable system theory.