Scientific machine learning (SciML) increasingly requires models that capture multimodal conditional uncertainty arising from ill-posed inverse problems, multistability, and chaotic dynamics. While recent work has favored highly expressive implicit generative models such as diffusion and flow-based methods, these approaches are often data-hungry, computationally costly, and misaligned with the structured solution spaces frequently found in scientific problems. We demonstrate that Mixture Density Networks (MDNs) provide a principled yet largely overlooked alternative for multimodal uncertainty quantification in SciML. As explicit parametric density estimators, MDNs impose an inductive bias tailored to low-dimensional, multimodal physics, enabling direct global allocation of probability mass across distinct solution branches. This structure delivers strong data efficiency, allowing reliable recovery of separated modes in regimes where scientific data is scarce. We formalize these insights through a unified probabilistic framework contrasting explicit and implicit distribution networks, and demonstrate empirically that MDNs achieve superior generalization, interpretability, and sample efficiency across a range of inverse, multistable, and chaotic scientific regression tasks.

翻译：科学机器学习（SciML）日益需要能够捕捉由不适定反问题、多稳态和混沌动力学引起的多模态条件不确定性的模型。尽管近期研究倾向于采用高度表达性的隐式生成模型，如基于扩散和流的方法，但这些方法通常数据需求量大、计算成本高，且与科学问题中常见的结构化解空间不匹配。我们证明，混合密度网络（MDNs）为SciML中的多模态不确定性量化提供了一种原则性但很大程度上被忽视的替代方案。作为显式参数化密度估计器，MDNs施加了一种针对低维、多模态物理量身定制的归纳偏置，能够直接全局分配不同解分支上的概率质量。这种结构带来了强大的数据效率，使得在科学数据稀缺的情况下也能可靠地恢复分离的模态。我们通过一个统一的概率框架对比显式和隐式分布网络，将这些见解形式化，并通过实证表明，在一系列反问题、多稳态和混沌科学回归任务中，MDNs实现了更优的泛化能力、可解释性和样本效率。