apply differential calculus

This work addresses the lack of a systematic approach to composing and ordering do-calculus rules, which hinders efficient exploration of the space of equivalent interventional queries. The paper introduces, for the first time, a derivation graph structure that formally captures the application and composition logic of do-calculus rules, systematically representing equivalence relations between observational and interventional probabilities under the do-calculus framework. Building upon this representation, the authors devise a streamlined identification procedure requiring at most four simplification steps. This approach not only reveals the intrinsic organizational structure underlying do-calculus reasoning but also enables the generation of multiple equivalent estimands for the same causal quantity, substantially improving estimation efficiency and facilitating practical applications of do-calculus.

Real-analytic differential-algebraic dynamic logic (DA-DL) lacks a sound proof calculus, hindering formal verification of high-index differential-algebraic equations (DAEs). Method: (1) Leveraging real-analytic function theory and index reduction, we establish a logically sound transformation of DAEs into equivalent ordinary differential equations (ODEs); (2) We introduce “ghost switching”, the first mechanism enabling exact decomposition of multimodal DAE systems into hybrid systems under precise conditions; (3) We design a DA-DL proof calculus compatible with the axiomatic foundation of differential dynamic logic (dL). Contribution/Results: Our framework guarantees logical equivalence throughout DAE transformations. We formally verify the Euclidean pendulum model, proving semantic equivalence between its original DAE formulation and the reduced ODE system. This work establishes a novel paradigm for modeling, reasoning about, and verifying high-index DAE systems—bridging a critical gap between symbolic DAE reduction and formal verification.

This study addresses the lack of a systematic framework for identifying critical input variables and conducting sensitivity analysis under uncertainty in complex simulations, particularly in military decision-making contexts. The authors propose a unified sensitivity analysis framework that integrates local and global methods—including variance-based, derivative-based, screening, and uncertainty quantification techniques—and strategically maps these approaches to specific decision objectives such as factor prioritization, fixing, variance reduction, and mapping. Innovatively, the framework introduces a “sensitivity audit” mechanism to enhance traceability of model assumptions and promote responsible model usage. By providing a structured guide for high-dimensional, complex simulation systems, this work significantly improves model interpretability, transparency, and the credibility of decisions derived from such models.

This paper addresses the lack of semantic foundations for automatic differentiation (AD) of algebraic data types (e.g., lists, trees) and arbitrary-order derivatives in higher-order functional languages. Methodologically, it introduces the first higher-order differentiable programming framework grounded in diffeological spaces, integrating categorical semantics and logical relations to formally define the semantics of forward-mode AD for higher-order functions and rigorously prove its structural preservation and Taylor-approximation completeness. Key contributions are: (1) the first complete correctness proof of AD semantics for languages with algebraic data types; (2) the first generalization of AD to arbitrary-order derivatives, unifying the behavior of derivative selection across primitive operations; and (3) the establishment of a unique macro-characterization of higher-order AD, providing a rigorous mathematical foundation for differentiable programming.

Traditional partial dependence plots (PDPs) yield distorted global sensitivity measures for engineering black-box models exhibiting strong input variable interactions. Method: This paper proposes a novel global sensitivity analysis framework based on individual conditional expectation (ICE) curves. It introduces a joint sensitivity metric combining ICE curve means and standard deviations—rigorously proven to lower-bound PDP-based sensitivity—and defines, for the first time, an ICE correlation metric quantifying the modulation strength of interaction effects on input–output relationships. The method integrates truncated orthogonal polynomial expansion with complementary interpretability tools (SHAP, Sobol’, PDP). Results: Evaluated on three canonical engineering benchmarks—a 5D analytical function, wind turbine fatigue modeling, and a 9D airfoil aerodynamic problem—the proposed approach consistently outperforms PDP, SHAP, and Sobol’ indices, delivering finer-grained insights into interaction effects and enhanced engineering interpretability via multi-perspective visualizations.

This work proposes an efficient numerical method for computing high-order Lie derivatives, which are essential in nonlinear system analysis yet computationally expensive when evaluated symbolically at increasing orders. By establishing a factorial-scaled equivalence between high-order Lie derivatives and the Taylor coefficients—computed along system trajectories and incorporating variational matrices—the approach circumvents symbolic manipulation entirely. The implementation leverages MATLAB’s ADTAYL package to automate differentiation and variational matrix computation, yielding substantial gains in computational efficiency. Experimental results on a gantry crane model demonstrate speedups of several orders of magnitude compared to MATLAB’s Symbolic Math Toolbox, highlighting the method’s practical advantage for high-order analyses in nonlinear control and dynamics.

This work addresses the challenge of preserving trajectory properties during simplification and transformation of hybrid systems involving differential-algebraic equations (DAEs). To this end, the paper introduces differential-algebraic refinement logic (dARL), a formal framework that builds upon trajectory semantics to support stepwise verification and simplification of DAE-based programs while guaranteeing semantic preservation at each transformation step. The core contribution lies in the first complete and provably correct refinement calculus for index reduction of DAEs, thereby establishing a formal foundation and syntactic assurance for incremental verification of complex DAE systems.

This study addresses the limitations of traditional network calculus, which assumes non-negative service curves and struggles to analyze complex systems with feedback control. By rigorously examining the properties of subadditive functions, the authors reveal that allowing negative service curves in feedback systems often leads to unstable analyses. To overcome this issue while preserving the non-negativity assumption, they develop a refined network calculus framework that integrates network calculus theory, subadditive function analysis, and system stability verification. Applying this approach to the complex feedback system proposed by Hamscher et al., the method achieves accurate modeling and tight performance bounds, effectively circumventing the instability inherent in prior techniques and significantly enhancing the applicability and reliability of network calculus in closed-loop systems.

This work addresses the structural identifiability of ordinary differential equation (ODE)-based mechanistic models—specifically, whether model parameters can be uniquely determined from ideal observational data—and proposes a unified symbolic analysis framework implemented in Julia. Built upon the StructuralIdentifiability.jl package, the framework integrates symbolic computation with parameter-output mapping analysis to support assessments of local and global identifiability, observability, and extraction of identifiable parameter combinations. As the first fully reproducible tutorial within the SciML ecosystem, it not only enables model reparameterization and informs experimental design but also demonstrates its efficacy across seven representative case studies spanning epidemiology, pharmacokinetics, and other domains, thereby offering both a practical workflow and theoretical foundation for modeling complex dynamical systems.