End-to-End Probabilistic Framework for Learning with Hard Constraints
Utkarsh Utkarsh, Danielle C. Maddix, Ruijun Ma, Michael W. Mahoney, Yuyang Wang
TL;DR
ProbHardE2E is a versatile end-to-end probabilistic framework that integrates hard constraints into neural network models, providing robust uncertainty estimates without distributional assumptions, applicable across diverse domains.
Contribution
It introduces a novel differentiable probabilistic projection layer enabling end-to-end learning with hard constraints in probabilistic models.
Findings
Effective in learning PDEs with uncertainty estimates
Improves probabilistic time-series forecasting accuracy
Supports complex non-linear constraints
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…
Peer Reviews
Decision·ICLR 2026 Poster
1. The main nolvety of this paper is to propose an end-to-end enforcement of hard constraints, where constraints are optimized as part of learning, not only enforced at post-processing or inference as in convention. 2. It also enables joint UQ and constraint satisfaction, thus suitable for safety/physics/constrained engineering tasks. 3. The sample-free & closed-form CRPS, combined with analytic projection propagations gives substantial training speedups. 4. Generality: the framework covers l
1. The major concern is that the method relies on first-order approximations to linearize the nonlinear function transformation, the KKT system, and the constraints, which together may bring too much estimation errors. Specifically, the covariance propagation relies on a first-order approximation of the Jacobian, thus the UQ under strong nonlinearity constraint can be misestimated. Also the linearized KKT system is only locally valid and can fail to converge or even diverge if the constraint is
- The method is clearly formulated in a principled “predictor–corrector” view. - The authors provide an exact handling for linear constraints. - The training objective is close-formed and sampling free.
- First-order DPPL approximation with no error bounds (risk under strong nonlinearity/variance). - Loss/evaluation assume independence while the projected posterior is generally correlated. - Practical experiments lock $Q$ to diagonal/identity, limiting the benefits of oblique projections and potentially biasing outcomes. - Nonlinear equality projection claims “strict feasibility,” but non-convexity and active-set issues are largely handled numerically without theoretical or robustness analys
1. The proposed method uses strictly proper scoring rules (e.g., CRPS) instead of log-likelihood objectives, reducing the learning bias caused by incorrect distributional assumptions. 2. The training process can be sample-free, offering potential efficiency advantages. 3. In both time-series forecasting and PDE-solving tasks, the proposed method demonstrates strong empirical performance.
1. During inference, the authors indicate that a projection must be computed at every step. Does this imply that the computational cost during inference could be substantial? Could the authors provide an analysis of the time complexity for both training and inference? Similarly, although the training process is sampling-free, repeatedly computing the Jacobian matrix can also increase computational time, especially when dealing with high-dimensional data or scenarios involving multiple constraint
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Taxonomy
TopicsAdversarial Robustness in Machine Learning · Gaussian Processes and Bayesian Inference · Model Reduction and Neural Networks