A Physics-Informed, Global-in-Time Neural Particle Method for the Spatially Homogeneous Landau Equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910049385840640 |
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| author | Kim, Minseok Son, Sung-Jun Kim, Yeoneung Lee, Donghyun |
| author_facet | Kim, Minseok Son, Sung-Jun Kim, Yeoneung Lee, Donghyun |
| contents | We propose a physics-informed neural particle method (PINN--PM) for the spatially homogeneous Landau equation. The method adopts a Lagrangian interacting-particle formulation and jointly parameterizes the time-dependent score and the characteristic flow map with neural networks. Instead of advancing particles through explicit time stepping, the Landau dynamics is enforced via a continuous-time residual defined along particle trajectories. This design removes time-discretization error and yields a mesh-free solver that can be queried at arbitrary times without sequential integration. We establish a rigorous stability analysis in an $L^2_v$ framework. The deviation between learned and exact characteristics is controlled by three interpretable sources: (i) score approximation error, (ii) empirical particle approximation error, and (iii) the physics residual of the neural flow. This trajectory estimate propagates to density reconstruction, where we derive an $L^2_v$ error bound for kernel density estimators combining classical bias--variance terms with a trajectory-induced contribution. Using Hyvarinen's identity, we further relate the oracle score-matching gap to the $L^2_v$ score error and show that the empirical loss concentrates at the Monte Carlo rate, yielding computable a posteriori accuracy certificates. Numerical experiments on analytical benchmarks, including the two- and three-dimensional BKW solutions, as well as reference-free configurations, demonstrate stable transport, preservation of macroscopic invariants, and competitive or improved accuracy compared with time-stepping score-based particle and blob methods while using significantly fewer particles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10874 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Physics-Informed, Global-in-Time Neural Particle Method for the Spatially Homogeneous Landau Equation Kim, Minseok Son, Sung-Jun Kim, Yeoneung Lee, Donghyun Numerical Analysis 82C40, 35Q83, 68T07 We propose a physics-informed neural particle method (PINN--PM) for the spatially homogeneous Landau equation. The method adopts a Lagrangian interacting-particle formulation and jointly parameterizes the time-dependent score and the characteristic flow map with neural networks. Instead of advancing particles through explicit time stepping, the Landau dynamics is enforced via a continuous-time residual defined along particle trajectories. This design removes time-discretization error and yields a mesh-free solver that can be queried at arbitrary times without sequential integration. We establish a rigorous stability analysis in an $L^2_v$ framework. The deviation between learned and exact characteristics is controlled by three interpretable sources: (i) score approximation error, (ii) empirical particle approximation error, and (iii) the physics residual of the neural flow. This trajectory estimate propagates to density reconstruction, where we derive an $L^2_v$ error bound for kernel density estimators combining classical bias--variance terms with a trajectory-induced contribution. Using Hyvarinen's identity, we further relate the oracle score-matching gap to the $L^2_v$ score error and show that the empirical loss concentrates at the Monte Carlo rate, yielding computable a posteriori accuracy certificates. Numerical experiments on analytical benchmarks, including the two- and three-dimensional BKW solutions, as well as reference-free configurations, demonstrate stable transport, preservation of macroscopic invariants, and competitive or improved accuracy compared with time-stepping score-based particle and blob methods while using significantly fewer particles. |
| title | A Physics-Informed, Global-in-Time Neural Particle Method for the Spatially Homogeneous Landau Equation |
| topic | Numerical Analysis 82C40, 35Q83, 68T07 |
| url | https://arxiv.org/abs/2603.10874 |