Parameter Interpolation Overview

• Parameter interpolation is a technique that estimates system behaviors at intermediate parameter values using refined function spaces and analytical frameworks.
• It underpins advances in PDE regularity, operator isomorphism, kernel methods, and model reduction, offering robust error analysis and computational efficiency.
• It plays a crucial role in data-driven applications, enabling smooth transitions in deep learning and uncertainty quantification by interpolating between discrete system states.

Parameter interpolation is a foundational analytical and computational strategy in mathematical analysis, numerical analysis, computational physics, machine learning, and engineering. The core objective of parameter interpolation is to estimate system behaviors, function values, or model states at intermediate values of a parameter—temporal, spatial, spectral, or model hyperparameter—by leveraging information at a finite set of parameter values. This enables finer resolution of physical effects, efficient bridging between regimes with distinct analytic representations, regularity theory in PDEs, model reduction for parameter-dependent systems, precise statistical estimation, robust optimization, and nuanced control or adaptation in data-driven and deep learning models. The following sections summarize principal methodologies, theoretical frameworks, and key application domains, drawing upon recent and foundational literature.

1. Interpolation Spaces and Function Parameters

The mathematical theory of interpolation of Hilbert and Banach spaces provides a rigorous framework for parameter interpolation in function spaces. Beyond traditional scales such as Sobolev spaces—indexed by a single real parameter (the order of smoothness)—research has established refined interpolation scales by introducing a function parameter in addition to the power parameter.

For example, the refined anisotropic Sobolev scale $H_2^{s,\varphi}(\mathbb{R}^2)$ consists of tempered distributions whose Fourier transforms satisfy

$$\int_{\mathbb{R}^2} (1 + |\xi|^2 + |\eta|^2)^s \varphi^2(\sqrt{1 + |\xi|^2 + |\eta|^2}) |\hat{w}(\xi, \eta)|^2\, d\xi\, d\eta < \infty$$

where $s\in\mathbb{R}$ is the main smoothness parameter and $\varphi$ is a positive function varying slowly at infinity in Karamata's sense (Los et al., 2013).

Crucially, these refined spaces are constructed as interpolation spaces between classical Sobolev spaces $H^{s_0}$ and $H^{s_1}$ using a function parameter $\psi$,

$H^{s,\varphi} = [H^{s_0}, H^{s_1}]_\psi$

with $\psi$ explicitly tied to $\varphi$. This methodology enables precise regularity calibration in settings too fine for classical integer- or real-indexed spaces, capturing "subpower" smoothness via slowly varying functions such as iterated logarithms, and plays a central role in advanced regularity theory for parabolic and elliptic PDEs (Anop et al., 2014).

2. Operator Interpolation and PDE Applications

Parameter interpolation is foundational in the theory of PDEs, particularly in the extension of classical results such as isomorphism theorems and a priori estimates for boundary value problems to general scales of function spaces.

For parameter-dependent families of elliptic or parabolic operators (for instance, operators depending on a spectral or physical parameter), one establishes uniform isomorphism results in refined and extended Sobolev (Hörmander) scales by interpolating known operator estimates for discrete parameter values. For instance, if

$$(A(x;\lambda), B(x;\lambda)) : H^\varphi(\Omega) \to H^\varphi(\Omega) \oplus \bigoplus_{j=1}^q H^{\varphi-m_j-1/2}(\Gamma)$$

is an isomorphism for all $|\lambda| \geq \lambda_0$ with constants independent of $\lambda$, then interpolation with respect to a function parameter enables generalization to extended smoothness indices and yields two-sided parameter-independent a priori estimates (Anop et al., 2014).

Furthermore, such interpolation directly underpins the transfer of well-posedness, regularity, and stability properties from classical Sobolev spaces to more nuanced scales, providing the analytical basis for improved solution theory in parabolic and parameter-elliptic PDEs (Los et al., 2013).

3. Kernel Methods and Data-Driven Parameter Interpolation

Kernel-based interpolation, particularly with radial basis functions (RBFs), is widely used for parameter interpolation in stochastic PDEs, machine learning, and scientific computing. The RBF metamodel approach constructs an interpolant

$$\tilde{u}(\zeta) = \sum_{j=1}^N \phi(\|\zeta - \zeta_j\|)\, c_j + P(\zeta)$$

where sample points $\zeta_j$ are selected in parameter space and the coefficients $c_j$ are solved to interpolate observed or simulated data (Steffes-lai et al., 2013).

To manage computational complexity in high-dimensional parameter spaces, techniques such as SVD-accelerated interpolation and parameter screening are employed. SVD reduces the effective rank, accelerating evaluation, while parameter screening (via local sensitives and Hessians) eliminates weakly influential parameters, alleviating the curse of dimensionality.

In the context of solving stochastic PDEs, this enables accurate and rapid surrogate modeling for uncertainty quantification, even when input distributions are not fully specified, and is computationally competitive with (and sometimes superior to) collocation methods (Steffes-lai et al., 2013).

Related advances feature improvements to RBF and Hermite RBF interpolation by introducing scaling terms (MHRBF) to counteract the ill-conditioning of infinitely smooth kernels (e.g., Gaussians) at low to moderate shape parameters. This provides robust, low-error interpolation for scattered data, essential for PDE solvers and scientific computation on irregular grids (Fashamiha et al., 21 Feb 2025).

4. Interpolation for Parameter-Dependent Model Reduction and System Identification

In parametric model order reduction and identification, interpolation is exploited directly in operator and trajectory spaces. For large parameter-dependent systems, the matrix inverse $A(\xi)^{-1}$ can be interpolated via a Frobenius-norm-projected linear combination of "snapshot" inverses: $$P_m(\xi) = \sum_{i=1}^m \lambda_i(\xi) P_i,\quad \text{with}\quad P_i = A(\xi_i)^{-1}$$ where $\lambda_i(\xi)$ are chosen via least-squares projection of the identity (Zahm et al., 2015).

For interpolating missing values in system trajectories governed by LPV systems, behavioral approaches represent admissible trajectories as linear combinations of columns of a Hankel matrix formed from measured data: $\tilde{w}_{[1,L]} = H_L(w)g$ and solve constrained linear systems matching known entries, ensuring consistency and uniqueness by fulfilling rank and nullspace conditions (Verhoek et al., 15 Aug 2025).

In model reduction of parametric bilinear systems, structure-preserving Petrov–Galerkin projections use interpolation to ensure that reduced subsystem transfer functions preserve both frequency and parameter dependencies of the original system, guaranteeing both function and sensitivity matching (Benner et al., 2020).

5. Parameter Interpolation in Statistical and Machine Learning Frameworks

Parameter interpolation enables efficient reconstruction, estimation, and generative modeling in both frequentist and Bayesian machine learning approaches, especially in high-dimensional models where direct computation is prohibitive.

For deep learning models, direct linear interpolation in weight space between fine-tuned model parameter vectors can yield controllable attribute changes (e.g., text sentiment) without significant performance degradation, as shown by the simple form

$$g_1(\alpha) = \alpha\, \theta_+ + (1-\alpha) \theta_-$$

and its generalizations involving pretrained initialization as anchoring (Rofin et al., 2022).

In molecular generation, parameter-space-based flow generative models (e.g., MolPIF) perform interpolation between a simple prior and data-target distribution parameters,

$$\theta_t = f(t)\, \theta_\text{data} + (1-f(t))\, \theta_\text{prior}$$

where the network is trained through KL-divergence-driven losses across this interpolation path, often outperforming sample-space flows in binding affinity, structural metrics, and stability (Jin et al., 18 Jul 2025).

Parameter interpolation methods are also integrated into adversarial training, where epoch-wise interpolation between past and present model parameters reduces oscillations and overfitting, enhancing robustness to adversarial attacks. This is performed by the update formula

$$\theta'_t = \lambda\, \theta'_{t-1} + (1-\lambda)\, \theta_t,$$

complemented by regularization aligning relative logit magnitudes (He et al., 2023).

6. Error Analysis and Anisotropy via Geometric Interpolation Parameters

Accurate analysis of interpolation error requires quantification of mesh anisotropy in finite element or geometric approximation. A geometric parameter,

$H_T = \frac{h_1 h_2 \cdots h_d}{|T|_d} h_T,$

combines edge lengths, element measures, and diameter to precisely capture anisotropy and mesh "shape quality" (Ishizaka, 22 Apr 2025, Ishizaka et al., 2021). Interpolation error bounds are then stated in the form

$$|\varphi - I_T\varphi|_{W^{m,q}(T)} \leq C \left(\frac{H_T}{h_T}\right)^m h_T^{\ell+1-m} |\varphi|_{W^{\ell+1, p}(T)}$$

with the uniform boundedness of $H_T/h_T$ equivalent to maximum-angle or shape-regularity conditions. This parameter enables sharp and explicit a priori estimates even for highly anisotropic elements, and is equivalent to classical measures like circumradius in two dimensions.

7. Practical Implications and Application Domains

Parameter interpolation is universally relevant wherever parameter-dependent models arise: regularity theory for evolutionary and parameter-elliptic PDEs, robust preconditioning across large parameter domains, model reduction in high-dimensional control systems, uncertainty quantification in engineering, high-resolution signal processing under compressive sampling (Fyhn et al., 2013), and data-driven system identification.

In specific applications, interpolation with function parameters substantiates advanced regularity scales for parabolic/elliptic PDEs (Los et al., 2013, Anop et al., 2014); Chebyshev interpolation in option pricing exploits analytic properties for sub/exponential error decay (Gaß et al., 2015); RBF/SVD interpolation enables efficient surrogate modeling for stochastic simulations (Steffes-lai et al., 2013); ergodic interpolation embeds discrete design spaces into continuous spaces for scalable system optimization (Karanjkar et al., 2014); and parameter interpolation in neural and generative models offers new directions for controllability and transfer in machine learning (Rofin et al., 2022, Jin et al., 18 Jul 2025).

In summary, parameter interpolation—in the generalized sense covering spaces, operators, systems, and data—serves as an organizing principle for methods that interpolate, estimate, and control system behaviors and function values across parameter spaces, with rigorous mathematical frameworks and diverse, high-impact real-world applications.