基于弱形式与帕累托子集选择的鲁棒数据驱动分数阶微分方程发现
Robust data-driven discovery of fractional differential equations via weak formulations and Pareto-based subset selection
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中文总结 AI 辅助
Weak-Pareto方法结合弱形式与帕累托子集选择，可从含噪数据中鲁棒发现分数阶微分方程，在多基准测试中表现优于无正则化方法及同期神经基线。
中文摘要 AI 辅助
分数阶偏微分方程描述非局部动力学，但从含噪数据中发现这类方程十分困难，因为分数阶微分会放大高频测量噪声，且导数阶数未知。我们提出Weak-Pareto方法，该方法将分数阶项的伴随一致性弱形式与离散项类型、连续分数阶阶数上的基于帕累托的子集选择相结合。对于线性右侧项，伴随算子将分数阶算子从测量场转移到光滑测试函数，用对噪声不敏感的平滑积分替代逐点微分；对于非线性项，其噪声抑制效果部分存在但仍有用。系数通过岭回归在分支感知差分进化搜索（用于确定阶数）中拟合，随后在验证误差-复杂度肘点处选择支持集大小。我们表明，固定线性右侧弱特征的方差会在网格细化时消失，而逐点分数阶特征中的噪声放大会随导数阶数增加而增大。在分数阶对流-扩散、反应-扩散及Burgers基准测试中，Weak-Pareto能从干净和含噪测量中恢复简约结构。在受控对流-扩散与Burgers对比中，它在所有 tested 乘性噪声水平下均保留正确支持集，而无正则化强形式对应方法一旦引入噪声便基本失效；该优势在加性高斯噪声下仍存在。消融实验显示，弱库驱动了噪声鲁棒性，且连续阶帕累托搜索避免了密集固定字典的支持集选择失败。在对流-扩散基准测试中，Weak-Pareto相比同期神经基线能产生更一致的算子恢复结果，且测量运行时显著更低。
英文摘要
Fractional partial differential equations describe nonlocal dynamics, but discovering them from noisy data is difficult because fractional differentiation amplifies high-frequency measurement noise and the derivative orders are unknown. We propose Weak-Pareto, which combines an adjoint-consistent weak formulation of fractional terms with Pareto-based subset selection over discrete term types and continuous fractional orders. For linear right-hand-side terms, the adjoint transfers fractional operators from measured fields to smooth test functions, replacing noise-sensitive pointwise differentiation with smoothing integration; for nonlinear terms, the noise-suppression effect is partial yet useful. Coefficients are fitted by ridge regression within a branch-aware differential-evolution search over the orders. The support size is then selected at the validation-error-complexity elbow. We show that the variance of fixed linear right-hand-side weak features vanishes under grid refinement, whereas noise amplification in pointwise fractional features increases with derivative order. Across fractional advection-diffusion, reaction-diffusion, and Burgers benchmarks, Weak-Pareto recovers parsimonious structures from clean and noisy measurements. In controlled advection-diffusion and Burgers comparisons, it retains the correct support at every tested multiplicative-noise level, whereas the unregularised strong-form counterpart largely fails once noise is introduced; this advantage persists under additive Gaussian noise. Ablations show that the weak library drives noise robustness and that continuous-order Pareto search avoids the support-selection failure of a dense fixed dictionary. On the advection-diffusion benchmark, Weak-Pareto yields more consistent operator recovery and substantially lower measured runtime than a contemporary neural baseline.
发表机构
- Center for Advanced Intelligence Project (AIP), RIKEN（理化学研究所先进智能项目中心（AIP））
- Graduate School of Information Science and Technology, The University of Osaka（大阪大学信息科学与技术研究生院）
机构由 AI 辅助整理，请以论文原文为准。
