A Multimodal Conditional Mixture Model With Distribution‐Level Physics Priors

ABSTRACT Many scientific and engineering systems exhibit intrinsically multimodal behavior arising from latent regime switching and non‐unique physical mechanisms. In such settings, learning the full conditional distribution of admissible outcomes in a physically consistent and interpretable manner remains a challenge. While recent advances in machine learning have enabled powerful multimodal generative modeling, their integration with physics‐constrained scientific modeling remains nontrivial, particularly when physical structure must be preserved or data are limited. This work develops a physics‐informed multimodal conditional modeling framework based on mixture density representations. Mixture density networks (MDNs) provide an explicit and interpretable parameterization of multimodal conditional distributions. We consider both Gaussian and Student's component distributions, the latter enabling robustness to heavy‐tailed variability. Physical knowledge is embedded through component‐specific regularization terms that penalize violations of governing equations or physical laws. This formulation naturally accommodates non‐uniqueness and stochasticity while remaining computationally efficient and amenable to conditioning on contextual inputs. The proposed framework is evaluated across a range of scientific problems in which multimodality arises from intrinsic physical mechanisms rather than observational noise, including bifurcation phenomena in nonlinear dynamical systems, stochastic differential equations (SDEs), and atomistic‐scale shock dynamics. In addition, the proposed method is compared with a conditional flow matching (CFM) model, a representative state‐of‐the‐art generative modeling approach, demonstrating that MDNs can achieve competitive performance while offering a simpler and more interpretable formulation.