AI 中文总结
本文提出Evo-GTransNet求解器，通过固定特征Galerkin方法线结合求积质量正交化解决抛物型偏微分方程，实现二阶时间收敛，在基准问题上的平均验证误差优于同类固定特征字典方法。
AI 中文摘要
本文开发了一种用于抛物型偏微分方程的进化型广义可迁移神经网络（Evo-GTransNet）求解器，其被构建为保留空间固定特征Galerkin方法线。GTransNet提供规定的空间字典，仅保留输出系数进行演化，从而避免了时间积分过程中的非线性训练。为解决严重的质量矩阵病态问题，我们应用求积加权截断奇异值分解（SVD）来选择数值可解的试验空间，随后进行单独的重缩放，使其基函数相对于组装求积质量内积正交归一化。秩截断修改了近似空间，而后续的正交归一化仅改变其坐标表示，并在精确算术运算中保留了保留的离散函数。所得的半离散系数系统具有单位质量矩阵，我们针对对称线性抛物型问题建立了半离散能量定律。结合隐式中点格式进行时间离散，我们进一步证明了该方法的收缩性，并推导了条件完全离散误差估计，其中误差由保留空间近似误差和相容性缺陷控制。数值实验表明，使用重复特征样本以及单独的组装和验证求积可实现二阶时间收敛，并量化了在存在严重原始质量病态情况下保留空间的准确性。对于本文考虑的高频和多尺度基准问题，在相同名义输出维度下，GTransNet在所有测试的固定特征字典中实现了最小的平均验证误差。
英文摘要
In this paper, we develop an evolutionary generalized transferable neural network (Evo-GTransNet) solver for parabolic partial differential equations, formulated as a retained-space fixed-feature Galerkin method of lines. A GTransNet provides the prescribed spatial dictionary, while only the retained output coefficients evolve, thereby avoiding nonlinear training during time integration. To address severe mass-matrix ill-conditioning, we apply a quadrature-weighted truncated singular value decomposition (SVD) to select the numerically resolved trial space, followed by a separate rescaling that makes its basis orthonormal with respect to the assembly-quadrature mass inner product. Rank truncation modifies the approximation space, whereas the subsequent orthonormalization changes only its coordinate representation and preserves the retained discrete functions in exact arithmetic. The resulting semidiscrete coefficient system has an identity mass matrix, and we establish a semidiscrete energy law for symmetric linear parabolic problems. With the implicit midpoint scheme for time discretization, we further prove the contractivity of the method and derive a conditional fully discrete error estimate in which the error is controlled by the retained-space approximation error and the consistency defects. Numerical experiments demonstrate second-order temporal convergence using repeated feature samples together with separate assembly and validation quadratures, and quantify the accuracy of the retained space in the presence of severe raw-mass ill-conditioning. For the high-frequency and multiscale benchmark problems considered here, GTransNet achieves the smallest mean validation errors among the tested fixed-feature dictionaries at the same nominal output dimension.
