mlsanityData to 14 Oct 2026

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

arXiv 2506.07003 ↗First posted 8 Jun 2025cs.LG cs.AI cs.CEcited by 9

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Abstract (arXiv)

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.

Key facts. Cited by 9 papers in the map (data through 14 Oct 2026). Builds on 61 of its 134 references that are in the map. First posted on arXiv 8 Jun 2025.

Builds on

61 of 134 references.

References · 61

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38 more
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50 most cited of 61 All 61 (JSON)

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 9.

9 papers.

  1. 2026Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewardscited by 2
  2. 2026Constraint-Aware Neurosymbolic Uncertainty Quantification with Bayesian Deep Learning for Scientific Discoverycited by 1
  3. 2026FluidGaussian: Propagating Simulation-Based Uncertainty Toward Functionally-Intelligent 3D Reconstructioncited by 0
  4. 2026Embedding Linear Equality Constraints in Probabilistic Neural Networks for Dynamic Modellingcited by 0
  5. 2026SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modelingcited by 0
  6. 2026End-to-End Learning of Safe Optimal Feedback Control in High Dimensions with Control Barrier Function Layerscited by 0
  7. 2026AT-SKM-Net: An Accelerated Trainable Sampling Kaczmarz-Motzkin Framework for Linear Hard-Constraint Feasibility on Dynamic Graphscited by 0
  8. 2026SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterizationcited by 0
  9. 2026LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problemscited by 0