$\partial$CBDs: Differentiable Causal Block Diagrams

2026-02-07

AI Summary

Existing modeling approaches struggle to simultaneously achieve compositionality, learnability, and verifiability in cyber-physical systems. This work proposes differentiable Causal Block Diagrams (∂CBDs), which unify the modular structure of causal block diagrams, assume-guarantee contract-based formal verification, and differentiable residual contracts compatible with automatic differentiation, thereby establishing an end-to-end optimization framework co-driven by data, physics, and constraints. By introducing trajectory-level differentiable residual contracts, the approach preserves formal guarantees throughout gradient-based optimization, realizing a novel modeling paradigm that seamlessly integrates compositionality, verifiability, and trainability. This significantly enhances the capability to efficiently learn and optimize cyber-physical systems under multi-source constraints.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Causal Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

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.

Problem

• cyber-physical systems
• causal block diagrams
• differentiable programming
• contract-based verification
• modularity

Innovation

• differentiable causal block diagrams
• assume-guarantee contracts
• residual-based contracts
• composable modeling
• automatic differentiation