$\partial$CBDs: Differentiable Causal Block Diagrams

TL;DR

This paper introduces $ ext{ extonehalf}CBDs$, a formalism that unifies causal block diagrams with differentiability and contract-based verification, enabling scalable, verifiable, and trainable modeling of cyber-physical systems.

Contribution

It presents a novel framework that combines modular causal modeling, assume--guarantee contracts, and differentiable certificates for end-to-end learning and verification.

Findings

Retains CBDs' modular structure and semantics

Incorporates differentiable residual-based contracts

Enables gradient-based optimization and verification

Abstract

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.