MiNO：用于偏微分方程的余切丛传播子学习

中文摘要 AI 辅助

针对偏微分方程的科学机器学习通常以解场（如物理信息神经网络）或解映射（如神经算子）为目标。本文研究第三种目标：传播子本身，即相空间中的相位与振幅。其动机在于正则性缺口：传输的间断在空间和时间上是非光滑的，但移动该间断的规则可以是携带单位振幅的多项式相位，因此生成演化的对象可能比其生成的场光滑得多。微局部神经算子（MiNO）学习该对象，利用程函方程确定相位、输运方程确定振幅，并通过振荡积分恢复解。由此，尖锐前沿与焦散线属于传播几何而非逐点拟合的场。小残差的意义超出了重构场：它们表明学习到的典范关系（携带奇异性的几何）接近精确的典范关系，且可区分可训练误差与频率截断尾。在匹配预算的间断平流基准测试中，MiNO在10000步内停止提升，达到其有限重构窗口的精度极限（该极限有闭式形式预测），而采用神经正切核损失平衡的物理信息神经网络则保持在初始误差附近；在光滑平流中，MiNO的平均误差为3.84×10⁻³，监督傅里叶神经算子的平均误差为3.12×10⁻²。单分支MiNO是对比中最小的模型，且一个训练好的生成器无需重新训练即可服务5个未见初始条件。

英文摘要

Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is $3.84\times10^{-3}$ for MiNO and $3.12\times10^{-2}$ for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.

发表机构

• Axiom Research Group（公理研究集团）
• The Nelson Mandela African Institution of Science and Technology（纳尔逊·曼德拉非洲科学技术研究院）
• African Institute for Mathematical Sciences（非洲数学科学研究所）
• The University of Tokyo（东京大学）

机构由 AI 辅助整理，请以论文原文为准。

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35 pages, 6 figures