arXiv:2511.06802cs.LG2025-11被引 10

用神经网络预估初始解,加速非线性有限元计算。

Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning

  • 神经算子先预测解的初始值,再用牛顿法修正。
  • 相比传统方法,计算量降低且精度保持稳定。
  • 适合需要快速模拟的大规模非线性力学问题。

我们提出一种基于神经算子初始化的牛顿法(NiN),用于加速计算固体力学中非线性问题的参数化求解。首先,训练一个物理信息约束的条件神经场,以近似控制方程的非线性参数解,建立参数空间与解空间之间的连续映射,可在任意空间分辨率下对给定参数进行评估。其次,由于神经算子预测可能存在误差,采用以神经输出为初值的牛顿法进行后续修正。通过对比三种策略:(i) 标准牛顿-拉夫森求解器(在NFEM中常用),虽鲁棒准确但计算成本高;(ii) 物理信息神经算子,推理快但训练分布外或高分辨率时精度下降;(iii) 神经初始化牛顿法(NiN),结合神经算子效率与NFEM鲁棒性。结果表明,该混合方法显著降低计算成本同时保持精度,展现出加速大规模非线性仿真应用的潜力。

原文摘要 · Abstract (English)

We propose a Newton-based scheme, initialized by neural operator predictions, to accelerate the parametric solution of nonlinear problems in computational solid mechanics. First, a physics informed conditional neural field is trained to approximate the nonlinear parametric solutionof the governing equations. This establishes a continuous mapping between the parameter and solution spaces, which can then be evaluated for a given parameter at any spatial resolution. Second, since the neural approximation may not be exact, it is subsequently refined using a Newton-based correction initialized by the neural output. To evaluate the effectiveness of this hybrid approach, we compare three solution strategies: (i) the standard Newton-Raphson solver used in NFEM, which is robust and accurate but computationally demanding; (ii) physics-informed neural operators, which provide rapid inference but may lose accuracy outside the training distribution and resolution; and (iii) the neural-initialized Newton (NiN) strategy, which combines the efficiency of neural operators with the robustness of NFEM. The results demonstrate that the proposed hybrid approach reduces computational cost while preserving accuracy, highlighting its potential to accelerate large-scale nonlinear simulations.

非线性有限元神经算子加速计算

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