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arXiv · 2506.07003

End-to-End Probabilistic Framework for Learning with Hard Constraints

Utkarsh Utkarsh·Danielle C. Maddix·Ruijun Ma·Michael W. Mahoney·Yuyang Wang
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Abstract

We present ProbHardE2E, a probabilistic forecasting framework that incorporates hard operational/physical constraints, and provides uncertainty quantification. Our methodology uses a novel differentiable probabilistic projection layer (DPPL) that can be combined with a wide range of neural network architectures. DPPL allows the model to learn the system in an end-to-end manner, compared to other approaches where constraints are satisfied either through a post-processing step or at inference. ProbHardE2E optimizes a strictly proper scoring rule, without making any distributional assumptions on the target, which enables it to obtain robust distributional estimates (in contrast to existing approaches that generally optimize likelihood-based objectives, which are heavily biased by their distributional assumptions and model choices); and it can incorporate a range of non-linear constraints (increasing the power of modeling and flexibility). We apply ProbHardE2E in learning partial differential equations with uncertainty estimates and to probabilistic time-series forecasting, showcasing it as a broadly applicable general framework that connects these seemingly disparate domains.

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Utkarsh Utkarsh, Danielle C. Maddix, Ruijun Ma, Michael W. Mahoney, Yuyang Wang. 2025-11-04. End-to-End Probabilistic Framework for Learning with Hard Constraints. https://arxiv.org/abs/2506.07003

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