Taylor-Expansion-Based Framework

• Taylor-Expansion-Based Framework is a method that represents global complexity by decomposing functions or systems into an infinite sum of locally computable derivative terms.
• It enables both deterministic and stochastic generalizations that drive high-order numerical methods, error analysis, and algorithmic acceleration in ODEs, PDEs, and simulations.
• The framework also extends to geometric analysis, statistical uncertainty quantification, and categorical formulations, bridging infinitesimal analysis with global structure.

The Taylor-Expansion-Based Framework is a class of methodologies, mathematical constructs, and categorical or stochastic generalizations that express an object (function, measure, process, or system response) locally in terms of its (possibly infinite) expansion in “derivatives” or generalized increments. At its core, the framework replaces the global complexity of a function or system with a (potentially infinite) sum of locally computable terms, each encoding information about the system’s response to infinitesimal variation. Taylor-expansion-based frameworks now appear across stochastic analysis, differential equations, probability (including random process expansions), differential geometry, numerical analysis, statistics, probability theory, and category theory, often serving as the primary bridge between infinitesimal analysis and global structure.

1. Analytic and Stochastic Generalizations

Classical Taylor expansions recast an analytic function $f$ in a neighborhood of $x_0$ as

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$

with convergence controlled by analyticity. This local expansion is generalized in several dimensions:

• Stochastic Taylor Expansions: For SDEs driven by Brownian or fractional Brownian motion, Taylor-like expansions represent the solution $X_t$ as

$X_t = x_0 + \sum_{k=1}^{\infty} g_k(t)$

where $g_k(t)$ is a sum over iterated integrals of the driving path, composed with repeated application of vector fields (Baudoin et al., 2011). The expansion is valid on a (possibly random) interval and convergence can be proven via fractional calculus and probabilistic L₂ estimates that exploit the regularity of the sample paths.

• Stochastic Taylor Expansions via Poisson Point Processes: Here, the Taylor expansion is stochastically generalized by representing the terms and order selection as a Poisson point process in coefficient-exponent space (Wu et al., 6 Aug 2025). The regression estimator becomes a random sum

$$\hat{f}(x) = \sum_{(a, n) \in \mathcal{N}} a (x-x_0)^{n}$$

where both the number of terms and the ordered set $\{(a,n)\}$ are random, enabling flexible, data-driven approximation and nonparametric regression.

• Taylor Measures: The Taylor expansion’s structure on the non-negative integers is internalized as a signed or probability measure

$$T_{(\gamma, \mathbf{a})}(B) = \sum_{n \in B} a_n \frac{\gamma^n}{n!}$$

for $B \subset \mathbb{N}$, bridging expansion theory, probability (discrete distributions), and Hilbert space geometry (Micheas, 6 Aug 2025).

2. Taylor Expansions in Differential Equations and Numerical Methods

Taylor-expansion-based frameworks provide foundational algorithms and convergence results for deterministic and stochastic differential equations:

• Multivariate ODEs and PDEs: High-order Taylor methods for ODE integration construct $u_{n+1}$ via truncated expansions of order $R$,

$$u(t_{n+1}) = u(t_n) + \sum_{\ell=0}^{R} \frac{h^{\ell}}{\ell!} u^{(\ell)}(t_n) + O(h^{R+1})$$

but require repeated symbolic differentiation. Approximate Taylor methods circumvent this by using centered finite difference operators to recursively estimate derivatives, yielding an arbitrarily high-order scheme without explicit symbolic differentiation and only requiring evaluations of $f$ (Baeza et al., 29 Jan 2025). These methods show competitive performance relative to classical Runge-Kutta at high order and in strongly coupled systems.

• Stochastic Taylor Expansions: For functionals of diffusion processes, repeated application of Itô’s formula generates a stochastic Taylor expansion whose terms capture both drift and Itô correction terms arising from nonzero quadratic variation (Rößler, 2013). Higher-order expansions are coded via colored rooted trees, generalizing the Butcher trees of deterministic ODE integration, and have direct implications for stochastic simulation and stochastic Runge–Kutta method design.

3. Geometric and Shape Analysis

• Shape Taylor Expansion: The response of wave fields (acoustic, electromagnetic) to domain perturbations is locally described by a Taylor expansion in terms of shape derivatives (Bao et al., 7 Jan 2025). High-order shape derivatives are computed using recurrence relations founded on exterior differential forms, material derivatives, and Lie derivatives. This yields a hierarchy of shape sensitivities crucial for inverse scattering, optimization, and uncertainty quantification.
• Geometry of Immersed Manifolds: On submanifolds, Taylor expansion of the exponential map captures global geometry via local invariants—second and third fundamental forms—encoding notions such as lateral and frontal deviation, asymptotic directions, and high-order contact with hyperspheres (Monera et al., 2012).

4. Probabilistic and Statistical Taylor Expansions

• Statistical Taylor Expansion: The classical expansion is extended by replacing deterministic input increments with independent random variables, yielding a direct formula for the mean and variance of $f(x+\tilde{x})$ in terms of distributional moments and derivatives (Wang, 2 Oct 2024). Statistical propagation of uncertainty can be implemented as “variance arithmetic,” providing error bounds and path-independence in complex computations under the uncorrelated uncertainty assumption.
• Probabilistic Taylor Expansion via Gaussian Processes: By defining “Taylor kernels” that are infinite series in powers of $(x-a)$, and conditioning on derivative data at a single point, the GP posterior mean exactly matches the Taylor polynomial (Karvonen et al., 2021). The RKHS structure and the diagonal character of the kernel under these observations translates into efficient, uncertainty-quantified local approximations.

5. Taylor Expansion in Category Theory and Abstract Settings

• Cartesian Differential Categories & Ultrametric Convergence: Objects such as maps $f: A \to B$ are equipped with higher-order derivatives via the axioms of a Cartesian differential category. The Taylor monomials $\mathcal{M}^{(n)}[f]$ form finite approximations $$\mathcal{T}^{(n)}[f] = \sum_{k=0}^n \mathcal{M}^{(k)}[f]$$, and a pseudo-ultrametric $d_D$ on homsets is defined by the least $n$ where the Taylor monomial differs, enforcing convergence of Taylor series to the map under this metric in “Taylor” categories (Lemay, 29 May 2024).
• Functorial and Monad-based Expansion: The generalization of the tangent bundle functor $T$ to higher order (“Taylor expansion functor” $T_n$) in a Cartesian differential category, together with combinatorial constructs for composition and naturality, yields a monad capturing the structure of higher-order dual numbers and “jets.” This underlies categorical treatments of automated differentiation (Walch, 13 Feb 2025).
• Bimonad for Summability: Extending summability to an infinitary functor $T$, equipped with bimonad structure and distributive laws with resource comonads, axiomatizes Taylor expansion in categories that may lack left additivity, thereby formalizing “analytic” behavior far beyond standard settings (Ehrhard et al., 2023).

6. Taylor-Expansion-Based Frameworks in Algorithms and Model Attribution

• Fast Algorithms and Kernel Expansions: Taylor expansion is foundational in algorithmic acceleration—e.g., in the TE-FMM for layered media Helmholtz equations, which compresses the Green’s function using Taylor expansions with recurrence and symmetry to achieve linear complexity and enable efficient multipole translations (Wanga et al., 2019).
• Adversarial Robustness in DNNs: The TEAM method approximates network output locally using a Taylor quadratic (including both gradient and Hessian), casting adversarial example generation as a constrained QP and integrating this as a tight inner problem in adversarial training (saddle point optimization) (Qian et al., 2020).
• Model Explanations: The Taylor expansion provides a rigorous, unified basis for local attributions in post-hoc explanation of opaque models (TaylorPODA) (Tang et al., 14 Jul 2025). Allocation postulates—precision, federation, zero-discrepancy—govern the share-out of expansion terms to features, while an “adaptation” property tunes explanation to align with empirical or task-specific criteria, yielding theory-backed attributions.

7. Measures and Statistical Structures

• Taylor Measures: By parameterizing measures as $$T_{(\gamma, \mathbf{a})}(B) = \sum_{n \in B} a_n \gamma^n / n!$$ (Micheas, 6 Aug 2025), building both deterministic and stochastic versions, the framework unifies the Taylor expansion view with discrete measure theory, probability distributions, Hilbert space structure, and the modeling of stochastic processes (including random walks and Brownian motion). Inner products, normalization, and the decomposition of positive and negative parts extend the reach of Taylor expansion into an extensive space of mathematical objects.

Tabular Overview of Central Axioms and Properties

Area	Key Taylor-Based Structure	Primary Application/Effect
Stochastic Analysis	Series of iterated integrals (Young, FP)	Expansion of SDE solution, error bound, convergence
Differential Geom.	Series in tangent vector	Geodesic deviation, local geometric invariants
Probability/Stat	Expansion via random terms/coefficients	Nonparametric regression, statistical inference
Category Theory	Functor/Monad on morphisms	Automated diff., semantic analytic constructs
Model Explanations	Taylor decomposition w/ allocation rules	Post-hoc attribution, theoretical explainability
Numerical ODE/PDE	Finite-diff. or recurrence Taylor approx	High-order schemes, algorithmic acceleration
Measure Theory	Signed/Probability Taylor measures	Unifying analytic/probabilistic structures

The Taylor-Expansion-Based Framework thus forms a central analytical, probabilistic, and categorical methodology, giving both structural understanding and algorithmic tools—ranging from stochastic differential equations to machine learning attribution, measure theory, and advanced categorical semantics—grounded in the expansion of objects via locally computable "derivatives" and their sum structures.