这篇论文展示了如何将科学原理嵌入AI训练过程，让模型能更准确地重建偏微分方程，对科学发现很有价值。
该研究提出了一种框架，通过将科学归纳偏置直接嵌入训练分布，学习可接受偏微分方程(PDEs)的连续潜在表示。实验结果表明，这种11维表示能准确重建广泛的代表性PDEs，并在方程家族内和家族间展现平滑的几何过渡。通过消融研究进一步证明，引入科学原理减少了方程形式的结构性误分类和参数估计误差。
Learning the Geometry of Admissible Hypotheses through Inductive Bias in Training Distributions
Scientific discovery often requires reasoning over competing hypotheses that are consistent with experimental observations. For mixed-variable and combinatorial hypothesis spaces, however, constructing probabilistic representations remains challenging because both the active model components and their associated parameters are unknown. In this work, we present a framework for learning continuous latent representations of admissible partial differential equations (PDEs) by embedding a scientific inductive bias directly into the training distribution. Progressively richer structural principles (e.g., sparsity, logical dependencies, common PDE families, and physical admissibility) are used to generate a structured distribution of hypotheses from which a gated variational autoencoder learns a continuous latent manifold. Experimental results show that the resulting 11-dimensional representation accurately reconstructs a broad collection of representative PDEs, while exhibiting smooth geometric transitions both within and across equation families. Through an ablation study we further demonstrate that introducing scientific principles reduces both structural misclassifications of equation forms and parameter estimation errors when reconstructing a representative benchmark set of admissible partial differential equations. These results show that embedding a scientific inductive bias in the training distribution enables the learning of compact and geometrically meaningful hypothesis manifolds, providing a principled foundation for future inference over competing governing equations.