通过分阶段加法修正，提升物理神经网络求解微分方程的稳定性与精度。
Variational Boosting for Physics-Informed Neural Networks
- 将解分解为多个小网络逐步修正，每步满足正交条件
- 小网络可全牛顿或共轭梯度优化，避免大模型病态问题
- 适用于需要高精度稳定求解非线性微分方程的研究者
物理信息神经网络（PINNs）通过最小化非线性算子残差来求解微分方程。然而，单体式PINNs常面临病态、谱偏倚和优化不稳问题。本文提出一种变分提升框架，解在函数空间中以加法方式构建。每一阶段训练一个弱学习器，其收敛修正满足局部正交性，等价于在神经网络函数流形切空间上的投影泛函梯度下降。由于每个修正网络规模小，受限最小化可采用全牛顿或共轭梯度更新，这在大型PINNs中通常不可行。该方法将全局非线性优化分解为一系列良定子问题，同时保持算子的完整变分结构。此框架为多阶段PINNs提供了几何解释，将其视为投影泛函梯度下降，并支持非线性微分方程的稳定二阶优化。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) solve differential equations by minimizing the residual of a nonlinear operator over a neural parameterization of the solution. However, monolithic PINNs often suffer from ill-conditioning, spectral bias, and optimization instability. We introduce a variational boosting framework in which solutions are constructed additively in function space. Each stage trains a weak learner whose converged correction satisfies a local orthogonality condition, equivalent to a projected functional gradient descent step onto the tangent space of the network's function manifold. Because each correction network is deliberately small, the restricted minimization admits full Newton or conjugate gradient updates, which are typically infeasible in large PINNs. The resulting method separates global nonlinear refinement into a sequence of well-conditioned subproblems while preserving the full variational structure of the operator. This framework provides a geometric interpretation of multi-stage PINNs as projected functional gradient descent and enables stable second-order optimization for nonlinear differential equations.
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