让因果块图可学习，同时保证系统正确性与模块化。
$\partial$CBDs: Differentiable Causal Block Diagrams
- 将因果块图与可微编程结合，支持梯度优化。
- 引入残差契约作为可微轨迹证书，实现端到端训练。
- 适合需要可验证、可训练的物理系统建模研究者。
现代网络物理系统（CPS）融合物理、计算与学习，亟需兼具可组合性、可学习性与可验证性的建模框架。现有方法各自为政：因果块图（CBDs）支持模块化连接但不可微；可微编程（DP）支持端到端梯度优化但缺乏正确性保障；合同式验证框架则与数据驱动模型优化脱节。为此，我们提出可微因果块图（∂CBDs），统一三者：(i) 保留CBDs的组合结构与执行语义；(ii) 引入假设-保证（A–G）合同进行模块化正确性推理；(iii) 提出基于残差的可微合同，作为与自动微分兼容的轨迹级证明，支持梯度优化与学习。三者协同构建了可扩展、可验证、可训练的建模流水线，兼顾因果性、模块化，并支持数据、物理与约束驱动的优化，适用于复杂CPS建模。
原文摘要 · Abstract (English)
Modern cyber-physical systems (CPS) integrate physics, computation, and learning, demanding modeling frameworks that are simultaneously composable, learnable, and verifiable. Yet existing approaches treat these goals in isolation: causal block diagrams (CBDs) support modular system interconnections but lack differentiability for learning; differentiable programming (DP) enables end-to-end gradient-based optimization but provides limited correctness guarantees; while contract-based verification frameworks remain largely disconnected from data-driven model refinement. To address these limitations, we introduce differentiable causal block diagrams ($\partial$CBDs), a unifying formalism that integrates these three perspectives. Our approach (i) retains the compositional structure and execution semantics of CBDs, (ii) incorporates assume--guarantee (A--G) contracts for modular correctness reasoning, and (iii) introduces residual-based contracts as differentiable, trajectory-level certificates compatible with automatic differentiation (AD), enabling gradient-based optimization and learning. Together, these elements enable a scalable, verifiable, and trainable modeling pipeline that preserves causality and modularity while supporting data-, physics-, and constraint-informed optimization for CPS.
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