Least-Squares Neural Network (LSNN) Method for Scalar Hyperbolic Partial Differential Equations
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| Format: | Preprint |
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2026
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| _version_ | 1866914285388562432 |
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| author | Liu, Min Cai, Zhiqiang |
| author_facet | Liu, Min Cai, Zhiqiang |
| contents | This chapter offers a comprehensive introduction to the least-squares neural network (LSNN) method introduced in [14,16], for solving scalar first-order hyperbolic partial differential equations, specifically linear advection-reaction equations and nonlinear hyperbolic conservation laws. The LSNN method is built on an equivalent least-squares formulation of the underlying problem on an admissible solution set that accommodates discontinuous solutions. It employs ReLU neural networks (in place of finite elements) as the approximating functions, uses a carefully designed physics-preserved numerical differentiation, and avoids penalization techniques such as artificial viscosity, entropy condition, and/or total variation. This approach captures shock features in the solution without oscillations or overshooting. Efficiently and reliably solving the resulting non-convex optimization problem posed by the LSNN method remains an open challenge. This chapter concludes with a brief discussion on application of the structure-guided Gauss-Newton (SgGN) method developed recently in [21] for solving shallow NN approximation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_20013 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Least-Squares Neural Network (LSNN) Method for Scalar Hyperbolic Partial Differential Equations Liu, Min Cai, Zhiqiang Numerical Analysis This chapter offers a comprehensive introduction to the least-squares neural network (LSNN) method introduced in [14,16], for solving scalar first-order hyperbolic partial differential equations, specifically linear advection-reaction equations and nonlinear hyperbolic conservation laws. The LSNN method is built on an equivalent least-squares formulation of the underlying problem on an admissible solution set that accommodates discontinuous solutions. It employs ReLU neural networks (in place of finite elements) as the approximating functions, uses a carefully designed physics-preserved numerical differentiation, and avoids penalization techniques such as artificial viscosity, entropy condition, and/or total variation. This approach captures shock features in the solution without oscillations or overshooting. Efficiently and reliably solving the resulting non-convex optimization problem posed by the LSNN method remains an open challenge. This chapter concludes with a brief discussion on application of the structure-guided Gauss-Newton (SgGN) method developed recently in [21] for solving shallow NN approximation. |
| title | Least-Squares Neural Network (LSNN) Method for Scalar Hyperbolic Partial Differential Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2601.20013 |