用于可解释算子学习的低秩柯尔莫哥洛夫-阿诺尔德生成器流
Low-Rank Kolmogorov-Arnold Generator Flows for Interpretable Operator Learning
AI总结：
KAFlow是一种低秩柯尔莫哥洛夫-阿诺尔德生成器流，可实现可解释算子学习，在合成基准、长时量子动力学等任务中性能优于现有方法，能发现稀疏回归无法得到的色散关系。
摘要：
从数据中学习物理定律的神经算子应能泛化到训练分布之外，并以闭式形式呈现所学定律。柯尔莫哥洛夫-阿诺尔德神经算子通过在所有位置和频率上学习稠密伪微分符号，实现了与位置相关动力学的上述特性，但其计算成本随分辨率的平方增长，且闭式形式需手动提取。该稠密符号并非必需：有限阶变系数微分算子的符号分离秩受限于其系数函数的数量，与分辨率无关，且该网络在自身基准测试中学习到的符号秩为2。我们提出柯尔莫哥洛夫-阿诺尔德流（KAFlow），其将物理算子的生成器学习为由单变量柯尔莫哥洛夫-阿诺尔德函数构成的低秩符号，固定这些函数的规范使每个项具有唯一位置，采用精确指数积分器积分动力学，并自动提炼其闭式形式。在柯尔莫哥洛夫-阿诺尔德神经算子的合成基准测试中，KAFlow将分布外损失降低3.4至11倍，无需符号模板即可将每个系数恢复至10^-4以内，达到数据的精度下限；在长时量子动力学任务中，其将状态不忠实度降低多达6个数量级，仅基于位置和动量测量训练时也降低30倍；KAFlow学习变阶分数算子的精度比现有算子网络高15倍，还能发现稀疏回归无法得到的非局域、分数阶及非多项式色散关系。
英文摘要：
Neural operators that learn physical laws from data should generalize beyond their training distribution and expose the learned law in closed form. The Kolmogorov-Arnold neural operator achieves both for position-dependent dynamics by learning a dense pseudo-differential symbol over all positions and frequencies, at a cost that grows with the square of the resolution and with closed forms that must be read out by hand. This dense symbol is unnecessary: a finite-order variable-coefficient differential operator has a symbol whose separation rank is bounded by the number of its coefficient functions, independently of the resolution, and the symbols that this network learns on its own benchmarks have rank two. We introduce the Kolmogorov-Arnold flow (KAFlow), which learns the generator of a physical operator as a low-rank symbol built from univariate Kolmogorov-Arnold functions, fixes the gauge of these functions so that every term has a unique place, integrates the dynamics with exact exponential integrators, and distills its closed form automatically. On the synthetic benchmarks of the Kolmogorov-Arnold neural operator, KAFlow lowers the out-of-distribution loss by 3.4 to 11 times and recovers every coefficient to within 10^-4 without symbolic templates, reaching the precision floor of the data. On long-horizon quantum dynamics, it lowers the state infidelity by up to 6 orders of magnitude, and by 30 times when trained only on position and momentum measurements. KAFlow also learns a variable-order fractional operator 15 times more accurately than existing operator networks and discovers nonlocal, fractional and non-polynomial dispersion relations on which sparse regression fails.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
