Derivative-Based Sensitivity Analysis

• Derivative-Based Sensitivity Analysis is a method that uses partial derivatives to quantify how small input changes affect model outputs.
• It offers computational efficiency and mathematically rigorous metrics for screening key factors and reducing model uncertainty.
• This approach complements variance-based indices and finds applications in engineering, geosciences, machine learning, and risk management.

Derivative-based sensitivity analysis is a class of global and local sensitivity analysis methodologies that quantify how small perturbations in model inputs propagate to changes in a model’s output, using (partial) derivatives of the output with respect to each input. This approach provides rigorous metrics for factor prioritization, model reduction, and uncertainty quantification—especially in high-dimensional, computationally demanding, or nonlinear systems. Derivative-based sensitivity measures (DGSMs) complement and sometimes substitute more expensive variance-based indices (such as Sobol’ indices) and are widely used in fields including uncertainty quantification, machine learning, inverse problems, risk management, and physical simulation.

1. Mathematical Foundations and Core Measures

Let $f(X)$ denote a real-valued model with input vector $X = (X_1, ..., X_d)$, defined on a probability space $(\Theta, \mu)$, where the $X_j$ are typically assumed independent with known distributions.

The primary derivative-based index for a scalar Quantity of Interest (QoI) is the expected squared derivative: $$\nu_j = \mathbb{E}\left[\left(\frac{\partial f}{\partial x_j}(X)\right)^2\right] = \int_\Theta \left(\frac{\partial f}{\partial x_j}(\theta)\right)^2 \mu(d\theta)$$ This is the classical DGSM. For vector- or functional-valued outputs, the DGSM generalizes to

$$\mathcal{V}_j(f; \mathcal{X}) = \int_\mathcal{X} \int_\Theta \left(\frac{\partial f}{\partial \theta_j}(s, \theta)\right)^2 \mu(d\theta) ds$$

where $f: \mathcal{X} \times \Theta \to \mathbb{R}$ (e.g., time, space, or other functional domains).

Several other formulations are prominent:

• The absolute derivative-based measure: $$\frac{1}{|\Theta|}\int_\Theta \left| \frac{\partial f}{\partial x_j}(\theta) \right| d\theta$$
• The raw (mean) derivative-based measure: $$\frac{1}{|\Theta|}\int_\Theta \frac{\partial f}{\partial x_j}(\theta) d\theta$$ (prone to cancellation and generally not robust for screening)
• Weighted or localized DGSMs: $$\int \left( \frac{\partial f}{\partial x_j}(x) \right)^2 w(x_j) d\mu(x)$$, with $w$ a bounded measurable weight.

For models with high-dimensional, functional, or correlated inputs, the corresponding measures involve integration over both input and functional domains, often leveraging Karhunen–Loève expansions for dimension reduction (Cleaves et al., 2019).

2. Theoretical Connections to Variance- and Entropy-Based Sensitivity Indices

Derivative-based sensitivity analysis is strongly linked to variance-based sensitivity analysis, in particular to Sobol’ indices. The total Sobol’ index $S_j^{\text{tot}}$ for parameter $X_j$ describes the share of total output variance attributable to $X_j$ (including all interactions): $S_j^{\text{tot}} = \frac{D_j^{\text{tot}}}{D}$ where $D_j^{\text{tot}}$ is the total partial variance with respect to $X_j$ and $D$ is the total variance.

A core result is that, for independent $X_j$ following probability law $\mu_j$ in the Boltzmann class (which includes uniform, Gaussian, and log-concave distributions), there exists a constant $C(\mu_j)$ such that

$D_j^{\mathrm{tot}} \leq C(\mu_j) \nu_j$

where $C(\mu_j) = 4C_1^2$, with $$C_1 = \sup_x \frac{\min[F_j(x), 1-F_j(x)]}{\rho_j(x)}$$ and $F_j$, $\rho_j$ the CDF and PDF, respectively (Lamboni et al., 2012). For log-concave measures, an alternative bound holds: $D_j^{\mathrm{tot}} \leq [\exp(v(m))]^2 \, \nu_j$ where $v$ is the potential in the Boltzmann density, and $m$ the median of $X_j$.

Recent advances extend these results by giving both lower and upper bounds on Sobol’ indices in terms of generalized and weighted DGSMs—including for nonuniform distributions—using Poincaré-type inequalities and Cheeger constants (Kucherenko et al., 2014, Kucherenko et al., 2016).

Additionally, derivative-based methods have been linked to entropy-based sensitivity indices. For a differentiable output $Y = g(X)$, the following holds: $$H(Y | X_{\sim i}) \leq H(X_i) + \mathbb{E}[\ln |\partial g / \partial x_i|]$$ Exponentiating this bound yields a normalized entropy proxy for total effect sensitivity that efficiently screens influential variables and aligns with classical proxies for linear Gaussian models (Yang, 2023).

3. Computational Strategies: Surrogates, Adjoint Methods, and High-Dimensional Models

A central motivation for DGSMs is scalability. Variance-based indices become prohibitive for high-dimensional or expensive simulations, because their computation requires a large number of model evaluations.

Key computational techniques include:

• Finite differences: Efficient for small or moderate-dimensional models but scales linearly with dimension.
• Automatic/Algorithmic Differentiation: Enables efficient and accurate computation of all required derivatives, especially in reverse mode for many parameters (cost roughly independent of parameter number) (Kucherenko et al., 2014).
• Adjoint-based methods: In models governed by PDEs or ODEs, sensitivity equations or adjoint formulations are solved alongside the forward problem. Sensitivities can be evaluated even when the number of inputs is very large, as the cost per parameter is low and often independent of the ambient parameter dimension (Cleaves et al., 2019, Chowdhary et al., 2023).
• Metamodeling/Surrogate models: Polynomial chaos expansions (PCE), Poincaré chaos expansions (PoinCE), or Gaussian process surrogates are fit using modest experimental designs. Once constructed, analytical differentiation of the surrogate provides closed-form DGSMs and Sobol’ indices at extremely low additional computational cost (Sudret et al., 2014, Lüthen et al., 2021, Belakaria et al., 13 Jul 2024).
• Active learning: Recent work applies targeted sequential experimental design to maximize information or variance reduction in the DGSM estimates, dramatically increasing efficiency for expensive black-box models (Belakaria et al., 13 Jul 2024).

In models with correlated input parameters, transformations such as Cholesky or Rosenblatt maps are used to transplant independent surrogate designs to the correlated input space, allowing derivative-based indices to capture both magnitude and sign changes due to correlation (Kardos et al., 2023).

The following table summarizes principal approaches for computing DGSMs and their use cases:

Computational Strategy	Model Type	Scalability/Cost
Finite Differences	General, low-dimension	Linear in $d$
Automatic Differentiation	General, moderate $d$	(Reverse mode) scalable
Adjoint Methods	ODE/PDE, high-dimension	Nearly independent of $d$
Surrogate Models (PCE, PoinCE, GP)	Black-box, high-cost	Low (after surrogate)
Active Learning	Expensive black-box	Minimal evaluations

4. Applications in Physical, Probabilistic, and Machine Learning Models

Derivative-based sensitivity analysis has been deployed across domains:

• Engineering and Physics: In thermal modeling of building envelopes, direct differentiation of the heat equation yields continuous sensitivity fields, identifying thermal conductivity as a dominant factor in thermal load uncertainty—at a cost 100 times lower than variance-based methods (Jumabekova et al., 2021).
• Geosciences and Biotransport: For PDE-constrained models, sensitivity of spatial/temporal fields is addressed by adjoint-based evaluation of KL-expansion coefficients, efficiently screening non-influential directions (Cleaves et al., 2019).
• Probabilistic Networks: Derivative-based local sensitivity in Bayesian and probabilistic networks supports robust parameter screening; fast, gradient-based methods such as the YODO algorithm scale to networks with hundreds of thousands of parameters (Gaag et al., 2013, Ballester-Ripoll et al., 2023).
• Machine Learning: DGSMs provide computationally tractable alternatives to expensive game-theoretic explainability techniques (Shapley values) and are extended to account for model uncertainty in Bayesian and deep learning models. The derivative-based Shapley value (DerSHAP) uses first- and second-order gradient information to assign feature importance at linear cost, outperforming KernelSHAP in runtime and in some cases in stability (Duan et al., 2023, Paananen et al., 2019).
• Stochastic Optimization and Risk Management: In problems where outputs are risk measures (e.g., quantiles, expected shortfall), derivatives may be defined with respect to infinitesimal distributional perturbations even when the input–output mapping is discontinuous (Pesenti et al., 2023).
• Generative Models: In modern diffusion models, closed-form, derivative-based sensitivity analysis predicts how score functions and generated samples shift under perturbations to the training distribution, enabling attribution and debugging without retraining (Scarvelis et al., 27 Sep 2025).

5. Advantages, Limitations, and Comparative Analysis

Major strengths of derivative-based sensitivity analyses include:

• Computational efficiency—permits high-dimensional screening with orders-of-magnitude fewer model evaluations than variance-based approaches (Lamboni et al., 2012, Sudret et al., 2014, Steenkiste et al., 2017).
• Mathematically rigorous bounds—DGSMs provide upper (and sometimes lower) bounds to Sobol’ indices, with the gap made precise via Poincaré and Cheeger inequalities (Kucherenko et al., 2014, Yang, 2023).
• Extensible to functional and correlated input cases—can be paired with dimension reduction (KL-expansions, active subspaces), surrogate modeling, and transformations for correlation (Cleaves et al., 2019, Kardos et al., 2023).
• Amenable to adjoint and automatic differentiation frameworks—enabling scalable evaluation for complex, structured models (Chowdhary et al., 2023, Steenkiste et al., 2017).

Limitations and caveats include:

• For highly nonlinear models with strong interaction effects or non-smooth mappings, derivative-based proxies (DGSM, entropy-DGSM) may rank inputs differently from variance-based (Sobol’/entropy) methods (Lamboni et al., 2012, Yang, 2023).
• For models with weak differentiability, discontinuity, or discrete variables, the traditional derivative-based approach requires adaptation, such as distributional derivatives or marginal quantile-based sensitivities (Pesenti et al., 2023).
• Theoretical DGSM–Sobol’ index bounds assume certain regularity conditions and independence among inputs; correlated settings require explicit adjustments (Kardos et al., 2023).

6. Extensions and Emerging Directions

Emerging research focuses on several axes:

• Bounding entropy-based indices: Recent work introduces derivative-based, computationally cheap upper bounds for conditional entropy-based sensitivity, widely applicable to non-Gaussian, skewed, or heavy-tailed uncertainties (Yang, 2023).
• Active learning for cost-effective estimation: Leveraging custom acquisition functions that target DGSM uncertainty, active learning frameworks drastically cut the number of evaluations required for DGSM estimation, critical for high-fidelity simulation and expensive experiments (Belakaria et al., 13 Jul 2024).
• Functional and infinite-dimensional problems: Low-rank and adjoint-based techniques enable derivative-based sensitivity for Bayesian inverse problems of PDE models, including global sensitivity and information gain quantification (Cleaves et al., 2019, Chowdhary et al., 2023).
• Data attribution and debugging in generative models: Closed-form methods predict the response of diffusion model outputs to infinitesimal data shifts, supporting model auditing and robustness analysis (Scarvelis et al., 27 Sep 2025).
• Combination of derivative-based and game-theoretic indices: New derivative-based Shapley values that naturally incorporate mixed partial derivatives provide tractable, fairness-axiomatized attribution in both global sensitivity and explainability domains (Duan et al., 2023).
• Handling discontinuities and quantile-based functionals: Differential quantile-based sensitivity frameworks extend classic derivative concepts to models with step changes or discrete risk drivers (Pesenti et al., 2023).

A plausible implication is that derivative-based sensitivity methods will remain central to scalable, precise, and interpretable analysis as model complexity, dimensionality, and data-richness increase—especially when paired with contemporary automatic differentiation, surrogate modeling, and active learning capabilities.