MuPS: Learning Multiple Physical Solution Branches from Unlabeled Outcomes

Learning-based physical simulation is typically built around an input–output map: given a physical condition, a model predicts a solution, or samples one from a conditional distribution. This formulation becomes inadequate when the underlying physical problem is intrinsically multi-solution, where the correct prediction target is a condition-dependent set of admissible solutions. In such settings, a single predictor may average incompatible solutions, while a probabilistic model may require many samples to reveal distinct or rare outcomes. We identify this as a fundamental limitation of current learning-based physics models and formulate the problem as learning a set-valued solution map from branch-unlabeled input–solution observations. We introduce MuPS, which augments existing physics-learning architectures with a small collection of explicit solution hypotheses and condition-dependent activation. The hypotheses provide direct access to distinct solution modes, while activation determines which hypotheses are admissible for each condition, allowing the represented solution-set cardinality to vary across the input domain. We demonstrate that this formulation applies to Fourier neural operators, neural fields, and conditional diffusion models. Across three physical and synthetic benchmarks and three model families, MuPS substantially improves best-of-K prediction accuracy on multi-solution problems, often by orders of magnitude, while providing more sample-efficient access to distinct solutions. It remains effective under strongly imbalanced solution observations and can retain comparable performance on a standard single-solution problem.