Scale-Adaptive Interpolation Schedules

• Scale-adaptive interpolation schedules are adaptive methods that replace fixed schemes with data-dependent refinement rules to allocate resolution where local error or sensitivity is highest.
• They integrate local error metrics, spectral properties, and optimization benefits to guide the placement of interpolation points in applications such as polynomial chaos and QAOA.
• These schedules have been applied in uncertainty quantification, molecular force prediction, and diffusion models to enhance accuracy and computational efficiency.

Searching arXiv for the cited papers to ground the article and verify identifiers. Scale-adaptive interpolation schedules are adaptive procedures for selecting interpolation coordinates, collocation nodes, basis functions, or time/noise parameters across heterogeneous problem scales such as polynomial degrees, spatial regimes, spectral bands, or circuit depths. Across the literature, the unifying idea is to replace fixed interpolation schedules with data- or model-dependent refinement rules that allocate resolution where local error, score mismatch, spectral roughness, optimization benefit, or physical stiffness is largest. In the context of adaptive sparse polynomial chaos expansions, the notion appears as a dimension-adaptive, anisotropic schedule over multi-indices and Leja nodes (Loukrezis et al., 2019). Closely related constructions arise in QAOA parameter interpolation (Apte et al., 2 Apr 2025), molecular force prediction (Yu, 8 Jun 2026), adaptive tempering in filtering (Rammelmüller et al., 2024), adiabatic state preparation (Guo et al., 11 Dec 2025), diffusion and stochastic-interpolant generative models (Williams et al., 2024, Chen et al., 3 Sep 2025, Chen et al., 1 Sep 2025, Damsholt et al., 3 Feb 2026), and adaptive interpolation for molecular potential energy surfaces (Kowalewski et al., 2016). This body of work suggests that “scale-adaptive interpolation schedules” are best understood not as a single algorithm, but as a design principle: schedules should be matched to the local geometry, sensitivity, and computational difficulty of the interpolation path.

1. Formal concept and recurring mathematical structure

A common formal pattern is the interpolation of a target object between two or more representations, together with an adaptive rule for deciding where to place the next interpolation point or how fast to traverse the path. In adaptive sparse polynomial chaos expansions, a scalar quantity of interest $u$ depending on $d$ independent random variables $\xi=(\xi_1,\dots,\xi_d)$ is approximated by a finite polynomial chaos expansion

$$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$

with multivariate orthonormal polynomials $\psi_\alpha$, and the schedule is the adaptive growth of the downward-closed multi-index set $\mathcal{I}$ together with its associated Leja collocation nodes (Loukrezis et al., 2019). In that setting, each refinement step adds exactly one node and one polynomial term, and the resulting expansion is interpolatory at the selected nodes.

In QAOA, the interpolated object is a smooth parameter schedule over normalized layer index $t=k/p\in[0,1]$, with

$$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$

where the schedule is adaptive in both circuit depth $p$ and basis dimension $m$ (Apte et al., 2 Apr 2025). In molecular force prediction, the interpolation coordinate is a scale variable $d$0 mediating between short-scale and long-range experts,

$d$1

with schedule refinement implemented through routing, differentiable updates, and scale-pool expansion (Yu, 8 Jun 2026).

In diffusion and stochastic interpolants, the interpolated object is typically a path of distributions. For denoising diffusion models, the schedule is a discretization of the diffusion path between $d$2 and $d$3, with adaptive time allocation derived from a local cost $d$4 or from spectral properties of the instance (Williams et al., 2024, Esteves et al., 19 Mar 2026). In stochastic interpolants, schedules are the scalar functions $d$5 in

$d$6

and scale adaptivity is imposed by controlling the Lipschitzness of the drift or the spectral conditioning of the interpolation (Chen et al., 3 Sep 2025, Chen et al., 1 Sep 2025, Damsholt et al., 3 Feb 2026).

These formulations differ in application domain, but they share three structural components. First, there is an interpolation family parameterized by time, degree, scale, or index. Second, there is a criterion measuring local difficulty, importance, or error. Third, there is an update rule that redistributes computational effort toward the scales where that criterion is largest. This suggests that the essential content of a scale-adaptive interpolation schedule is the coupling of interpolation geometry with problem-dependent refinement indicators.

2. Adaptive sparse interpolation and polynomial chaos

The 2019 Leja-interpolation construction provides a canonical example of a scale-adaptive interpolation schedule in uncertainty quantification (Loukrezis et al., 2019). The method starts from stochastic collocation on Leja nodes and exploits a hierarchical Newton-like basis whose multivariate degrees are unique. The interpolating polynomial chaos expansion is obtained either through an explicit hierarchical basis transform or directly in the orthonormal basis, because each Leja node is in one-to-one correspondence with a basis polynomial degree.

The univariate Leja sequence is defined recursively by

$d$7

with $d$8 a positive weight linked to the orthogonality measure. For polynomial chaos with respect to $d$9, the paper states that a common and effective choice is $\xi=(\xi_1,\dots,\xi_d)$0, and this empirically improves stability and adaptivity across distributions including uniform, Gaussian, truncated, and Gumbel cases (Loukrezis et al., 2019). Because Leja sequences are nested, new points append to the previous set rather than replacing it.

The adaptive schedule is organized over a downward-closed multi-index set $\xi=(\xi_1,\dots,\xi_d)$1, satisfying

$\xi=(\xi_1,\dots,\xi_d)$2

Downward-closedness ensures hierarchical consistency and sparse-grid structure. Refinement occurs on the admissible frontier using indicators that measure contribution at different scales. The data block lists two examples: $\xi=(\xi_1,\dots,\xi_d)$3 The former is surplus-based in the hierarchical Newton basis; the latter is the orthonormal polynomial variance contribution.

Anisotropy enters through weights $\xi=(\xi_1,\dots,\xi_d)$4, with priority score

$\xi=(\xi_1,\dots,\xi_d)$5

where $\xi=(\xi_1,\dots,\xi_d)$6 is monotone. The resulting schedule is therefore “scale-adaptive” in two senses: it resolves polynomial degree and physical dimension simultaneously, and it can preferentially refine influential dimensions. The procedure begins with $\xi=(\xi_1,\dots,\xi_d)$7, constructs the admissible frontier, computes indicators, selects the index maximizing priority, appends the corresponding Leja node, maintains downward-closedness, and terminates when a global criterion such as frontier-indicator decay, simulation budget, or cross-validation stabilization is reached (Loukrezis et al., 2019).

Two properties are central. First, each collocation point adds exactly one polynomial term, yielding “one polynomial term per collocation point.” Second, the resulting PCE is exact on the collocation nodes,

$\xi=(\xi_1,\dots,\xi_d)$8

This distinguishes the schedule from pseudo-spectral projection and regression-based collocation, which often require oversampling. A plausible implication is that scale-adaptive interpolation schedules are especially natural when the approximation architecture supports a one-to-one relation between refinement units and basis terms.

3. Representation-capacity schedules in optimization and control

A second major family of scale-adaptive interpolation schedules appears in optimization of parameterized trajectories. In QAOA, the schedule is the ordered set of $\xi=(\xi_1,\dots,\xi_d)$9 parameters $$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$0, but the iterative interpolation method replaces direct optimization over all $$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$1 angles by optimization over $$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$2 basis coefficients with $$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$3 (Apte et al., 2 Apr 2025). The construction treats the layer angles as smooth functions of $$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$4, expands them in orthogonal-function bases such as Chebyshev, Legendre, or Fourier, and discretizes by

$$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$5

The adaptive schedule is not only over depth $$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$6, but jointly over depth $$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$7 and representation capacity $$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$8. The algorithm begins from small $$u(\xi) \approx \sum_{\alpha\in\mathcal{I}} c_\alpha \psi_\alpha(\xi),$$9 and $\psi_\alpha$0, optimizes the $\psi_\alpha$1 coefficients, measures relative performance improvement

$\psi_\alpha$2

and increases $\psi_\alpha$3 only after $\psi_\alpha$4 consecutive rounds with $\psi_\alpha$5 (Apte et al., 2 Apr 2025). Depth is increased each round, with warm-starting from the previous coefficients, and newly added basis modes are initialized at zero. The paper reports that this reaches schedules exceeding 1000 layers on LABS instances and yields better performance with fewer optimization steps than current approaches on SK, portfolio optimization, and LABS (Apte et al., 2 Apr 2025).

This schedule is “scale-adaptive” because it allocates optimization capacity in accordance with the observed smoothness of the optimal schedule. The paper notes that coefficient spectra decay rapidly, and that reconstructing a $\psi_\alpha$6 schedule with only approximately the first 12 coefficients yields similar performance. This suggests that the adaptive mechanism is exploiting a low-dimensional manifold structure in high-depth schedules rather than merely compressing parameters.

Adiabatic state preparation supplies a related but physically distinct notion of schedule adaptivity. There the schedule is a strictly increasing function $\psi_\alpha$7 in

$\psi_\alpha$8

and the adaptive rule slows the evolution in small-gap regions and speeds it up elsewhere (Guo et al., 11 Dec 2025). The paper introduces the power-law ODE

$\psi_\alpha$9

with

$\mathcal{I}$0

Under the spectral gap measure condition

$\mathcal{I}$1

this improves the runtime scaling for constant accuracy from $\mathcal{I}$2 under linear scheduling to $\mathcal{I}$3 (Guo et al., 11 Dec 2025). For linear or piecewise linear gaps, $\mathcal{I}$4 satisfies the Euler–Lagrange optimality condition for the derived adiabatic-error functional. The paper also states that the linear schedule is never optimal unless the gap is constant.

Both QAOA and adiabatic scheduling therefore instantiate the same principle in different variables: refinement should follow the latent complexity of the path. In one case, the complexity is empirical schedule bandwidth; in the other, it is instantaneous spectral gap.

4. Routing, gating, and discrete scale-pool refinement

In multiscale representation learning, scale-adaptive interpolation schedules often appear as mixtures of experts with learnable interpolation weights. The molecular force-prediction framework on a NaCl aqueous ionic system is explicit about this interpretation (Yu, 8 Jun 2026). It treats two predefined scales as initial anchors, $\mathcal{I}$5, where $\mathcal{I}$6 denotes the long-range expert and $\mathcal{I}$7 denotes the short-scale expert. The prediction for atom $\mathcal{I}$8 is

$\mathcal{I}$9

or, in residual form,

$t=k/p\in[0,1]$0

The schedule variable $t=k/p\in[0,1]$1 is learned or optimized under the component-wise force MAE

$t=k/p\in[0,1]$2

The paper derives the subgradient with respect to $t=k/p\in[0,1]$3,

$t=k/p\in[0,1]$4

and uses this to update either gate parameters or the interpolation weights directly (Yu, 8 Jun 2026).

A distinctive contribution is scale-pool refinement. Starting from endpoint anchors $t=k/p\in[0,1]$5, the method defines interpolated experts between anchor pairs,

$t=k/p\in[0,1]$6

chooses $t=k/p\in[0,1]$7 minimizing the task loss, and inserts the resulting scale if the improvement $t=k/p\in[0,1]$8 exceeds a threshold $t=k/p\in[0,1]$9 (Yu, 8 Jun 2026). The reported update trajectory is

$$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$0

followed by insertion of $$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$1 and $$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$2, yielding the final scale pool

$$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$3

This final updated pool achieves an overall MAE of $$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$4, compared with $$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$5 for the long-only baseline, $$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$6 for oracle hard routing, and $$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$7 for continuous oracle interpolation (Yu, 8 Jun 2026). In the close-contact regime with nearest-ion distance below $$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$8 nm, the close-contact MAE decreases from $$\gamma(t)=\sum_{i=1}^{m} c_i^{(\gamma)} \phi_i(t), \qquad \beta(t)=\sum_{i=1}^{m} c_i^{(\beta)} \phi_i(t),$$9 to $p$0 under continuous oracle interpolation, and to $p$1 under the final scale-pool update.

The main technical significance is that the scale schedule is discovered rather than fixed. Endpoint anchors dominate globally, but intermediate scales become important in difficult transitional regimes. The reported anchor-usage statistics quantify this: over all atoms, $p$2 accounts for $p$3, $p$4 for $p$5, and all intermediates for $p$6; in the close-contact regime, intermediates rise to $p$7, with $p$8 increasing to $p$9 (Yu, 8 Jun 2026). This directly supports the claim that adaptive schedules are most valuable where regime boundaries are sharp but not binary.

5. Path discretization, tempering, and generative schedules

A large and mathematically diverse literature studies scale-adaptive interpolation schedules for paths of probability measures. In adaptive tempering for filtering, the canonical path is

$m$0

and the schedule is the choice of intermediate tempering levels (Rammelmüller et al., 2024). The paper introduces a finite schedule with $m$1 steps, $m$2, so that the likelihood is split into $m$3 and $m$4 components, and different filters are applied to different substeps. Adaptation is driven either by effective sample size,

$m$5

or by an interquartile-range criterion in observation space. If one chooses to adapt $m$6, the step is selected so that $m$7, using bisection or line search (Rammelmüller et al., 2024). The resulting schedule is explicitly scale-sensitive to likelihood curvature, dimensionality, prior spread, and observation mismatch.

Annealed importance sampling provides another path-based formulation. There, a geometric path

$m$8

is paired with a constant-rate discretization schedule based on local divergence curvature (Goshtasbpour et al., 2023). For the geometric path, the local KL approximation yields

$m$9

hence

$d$00

This is a direct example of scale-adaptive interpolation scheduling: smaller steps are allocated where the log-density difference has larger variance.

Diffusion models yield yet another version. In “Score-Optimal Diffusion Schedules,” the adaptive object is the discretization schedule along the reverse diffusion path, and the local cost is defined as

$d$01

with the corrector-optimized version

$d$02

The paper proves a dense-limit geometry with local metric density $d$03 and shows that the optimal schedule equalizes the cumulative length

$d$04

leading to the constant-speed geodesic allocation

$d$05

after interpolation of empirical cumulative costs (Williams et al., 2024). The method is hyperparameter-free in its core UpdateSchedule routine and uses only score evaluations. On CIFAR-10 with NFE 35 and Heun solver, the paper