发表机构
Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究针对物理信息神经网络求解微分方程存在的问题，提出变分提升框架，通过构建加法解、训练弱学习者满足局部正交性条件，实现全局非线性细化分离为条件良好子问题，提供几何解释并实现稳定二阶优化。
AI 中文摘要
物理信息神经网络（PINNs）通过在解的神经参数化上最小化非线性算子的残差来求解微分方程。然而，整体式PINNs常常存在病态、频谱偏差和优化不稳定性问题。我们引入了一个变分提升框架，其中解在函数空间中以加法方式构建。每个阶段训练一个弱学习者，其收敛校正满足局部正交性条件，等同于在网络函数流形的切空间上进行投影函数梯度下降步骤。由于每个校正网络刻意设计得小，受限最小化允许进行完整的牛顿或共轭梯度更新，这在大型PINNs中通常不可行。所得方法将全局非线性细化分离为一系列条件良好的子问题，同时保留算子的完整变分结构。该框架为多阶段PINNs提供了作为投影函数梯度下降的几何解释，并实现了非线性微分方程的稳定二阶优化。
英文摘要
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.
