发表机构
School of Mechanical and Material Engineering; University College Dublin; Massachusets Institute of Technology（机械与材料工程学院; 都柏林大学学院; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过将科学归纳偏置嵌入训练分布，用门控变分自编码器学习可容许PDE的连续潜在流形，实现了11维表示对PDE的准确重构，减少了方程误分类和参数估计误差。
AI 中文摘要
科学发现通常需要对与实验观测一致的竞争性假设进行推理。然而，对于混合变量和组合假设空间，构建概率表示仍具挑战性，因为主动模型组件及其相关参数均为未知。本研究提出一种框架，通过将科学归纳偏置直接嵌入训练分布，学习可容许偏微分方程（PDE）的连续潜在表示。采用逐步丰富的结构原则（如稀疏性、逻辑依赖关系、常见PDE族及物理可容许性）生成结构化假设分布，门控变分自编码器（gated variational autoencoder）从该分布中学习连续潜在流形。实验结果表明，所得11维表示可准确重构大量代表性PDE，且在方程族内部及之间呈现平滑几何过渡。通过消融研究进一步证明，引入科学原则可减少方程形式的结构误分类，以及重构代表性基准可容许偏微分方程集时的参数估计误差。这些结果表明，在训练分布中嵌入科学归纳偏置可实现紧凑且具有几何意义的假设流形学习，为未来对竞争性控制方程的推理提供了有原则的基础。
英文摘要
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.
