让神经网络自动满足物理约束并给出可信度估计。
End-to-End Probabilistic Framework for Learning with Hard Constraints
- 用可微投影层直接嵌入硬约束，端到端训练。
- 无需假设分布，能准确估计不确定性，优于传统方法。
- 适用性强，可用于微分方程与时间序列的不确定性建模。
我们提出ProbHardE2E，一种将硬性运筹/物理约束融入概率预测的框架，同时提供不确定性量化。该方法采用新型可微概率投影层（DPPL），可与多种神经网络架构结合。与需后处理或推理时满足约束的现有方法不同，DPPL支持端到端学习。模型优化严格合理的评分规则，不依赖目标分布假设，从而获得更鲁棒的分布估计（相较依赖似然的目标函数，后者易受分布假设和模型选择偏差影响）；且可处理多种非线性约束，提升建模能力与灵活性。我们在带不确定性的偏微分方程学习及概率时间序列预测中应用该框架，证明其在看似无关领域间具有广泛适用性。
原文摘要 · Abstract (English)
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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