从微分方程数据中自动发现李点对称性的新方法
Data-driven discovery of intrinsic Lie point symmetries of differential equation systems
一篇偏数学的方法论文：不用推导符号方程，直接从微分方程数据里挖出连续对称性生成元，在 11 维基准上全部恢复，做科学机器学习的人可以看看。
arXiv 论文提出将李点对称性发现转化为采样 jet 空间数据上的算子理论问题，无需符号形式的决定方程或封闭形式的控制方程。框架结合延拓、逐次子空间约束和对可训练系数线性的非线性字典，并用 rank-stabilization 预检测试诊断对称代数不稳定的系统。在一个已知具有 11 维表示内蕴对称代数的受控微分方程基准上，该方法完整恢复了表示生成元集合，并以数值精度重现其李括号结构，还能将已知生成元近似外推到训练域之外。
Data-driven discovery of intrinsic Lie point symmetries of differential equation systems
Discovering continuous symmetries directly from differential-equation data can reveal geometric structure for model reduction and scientific machine learning without symbolic determining equations or a closed-form representation of the governing system. We formulate intrinsic Lie point-symmetry discovery as an operator-theoretic problem on sampled jet-space data. Admissible generators are identified from a sampled determining operator and separated from redundant, solution-dependent, and geometrically trivial directions. The framework combines prolongation, sequential subspace restrictions, and a fixed nonlinear dictionary linear in trainable coefficients. A rank-stabilization preflight test diagnoses systems whose represented intrinsic symmetry algebra does not stabilize under dictionary refinement. On a controlled differential-equation benchmark with an analytically known 11-dimensional represented intrinsic symmetry algebra, the method recovers the complete represented generator set, reproduces its Lie-bracket structure to numerical precision, and approximates a known generator beyond the training domain. Here we show that explicit intrinsic continuous-symmetry generators can be recovered from data obtained from exact or approximate differential systems.