System Interpolation: Multi-Domain Approaches

• System interpolation is a framework where interpolation is performed on structured systems rather than isolated functions, influencing overall system behavior.
• It spans diverse applications including nonuniform sampled-data reconstruction, frequency-domain model reduction, PDE regularity, and parameter-space tuning for language models.
• Practical implementations demonstrate significant efficiency gains and improved accuracy in hardware architectures, control systems, and adaptive deep learning configurations.

System interpolation denotes a family of constructions in which interpolation is performed at the level of a system rather than merely at the level of a scalar function. In the cited literature, the term ranges from nonuniform sampled-data reconstruction with Chebyshev nodes and word-serial hardware architectures, to frequency-domain interpolation of transfer functions for model reduction and identification, to direct completion of missing LPV trajectories from data, to interpolation inequalities used to close nonlinear PDE estimates, to interpolation-derived integrable systems, to proof-theoretic effective interpolation, and to parameter-space interpolation between language-model checkpoints (Tulabandhula, 2010, Jonas et al., 3 Jun 2025, Verhoek et al., 15 Aug 2025, Degtyarev, 2024, Yamada, 2017, Maxa, 6 Jan 2026, Yang et al., 29 Jan 2026). The unifying feature is that interpolation acts on a structured object—signals, trajectories, transfer functions, solution classes, proof predicates, or checkpoint parameters—and thereby changes the behavior of an entire system.

1. Conceptual range of the term

The literature uses “system interpolation” in several technically distinct but structurally related senses.

Setting	Interpolated object	System-level role
Sampled-data and sensing	Signals, soundfields, virtual views	Reconstruction, hardware realization, remote sensing
Dynamical systems	Transfer functions, tangential data, spectral zeros	Model reduction, identification, passivity preservation
Behavioral/data-driven systems	Finite trajectories under scheduling	Missing-data completion without explicit model identification
PDE and weak-solution theory	Intermediate norms or solution regimes	Closing estimates or interpolating solution behavior
Integrable and logical systems	Determinant data, proof obligations	Generating dynamics or extracting separators/proofs
Parameter-space model control	Checkpoint parameters	Continuous control of reasoning intensity

In sampled-data settings, interpolation typically reconstructs a signal or field from sparse or nonuniform observations. In control and model reduction, interpolation is usually frequency-domain and transfer-function based, with exact matching at selected frequencies or directions. In behavioral system theory, interpolation can mean completing a partially specified trajectory so that it belongs to the admissible behavior. In PDE analysis, interpolation often refers to inequalities between summability and smoothness spaces, or to a continuous parameter between conservative and dissipative solution concepts. In integrable systems and logic, interpolation is not primarily approximation; it is the device from which the system itself is derived or the proof-theoretic property is extracted (Doliwa, 2022, Maxa, 6 Jan 2026).

This diversity suggests that “system interpolation” is best understood as an umbrella term for constructions where interpolation is constrained by, or acts directly upon, an ambient system structure rather than an isolated function value table.

2. Sampled-data, acoustic, and viewpoint systems

A hardware-oriented meaning of system interpolation appears in the Chebyshev sampled-data setting of “Some Architectures for Chebyshev Interpolation” (Tulabandhula, 2010). The signal is sampled over $x\in[-1,1]$, but not on an equispaced grid; instead, the sample locations are the roots of $T_{N+1}$,

$$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$

The reconstruction polynomial is written in a Chebyshev basis,

$P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$

with coefficients obtained by a DCT-like transform. The paper contrasts transform-domain and time-domain realizations, introduces Chebyshev Type Interpolation Functions $\phi_i(x)$, and then proposes a word-serial systolic-array architecture for an 8-sample window. Its first stage computes $\{c_i\}$, its second stage evaluates Chebyshev polynomials recursively using a reused multiplier-adder pair, and its output stage performs an FIR-like accumulation. The dependence-graph scheduling is summarized by

$T=[1\;\;0], \qquad s=[1\;\;1]^T.$

For $N=8$, the reported latency is 16 cycles, with 100% hardware usage efficiency. The same paper also links interpolation to the sampling front end: for the bandlimited test signal $\sin(4x)+0.5\sin(8x)$, error $<1.1\%$ required 10 equispaced points but 8 Chebyshev points; for $T_{N+1}$0, error $T_{N+1}$1 required 11 equispaced points but 8 Chebyshev points. Under an 8-bit flash/SAR comparison-count model, the resulting Flash-SAR hybrid consumed 1552 au per window versus 2560 au and 2816 au for the equispaced flash-only alternatives, corresponding to approximately 39% and 44% power savings.

In active noise control, interpolation is used to synthesize virtual error microphones inside a region of interest from physical microphones outside it (Zhang et al., 2023). The proposed system is a simulated 3D free-field ANC setup with one primary source, two secondary sources, eight monitoring microphones, and two virtual ear microphones. The interpolated field $T_{N+1}$2 is produced by a PINN constrained by the homogeneous wave equation,

$T_{N+1}$3

through a loss

$T_{N+1}$4

The network has one hidden layer with 16 neurons, $T_{N+1}$5 activation, $T_{N+1}$6 collocation points, and $T_{N+1}$7 training epochs. Relative to a spherical-harmonic baseline, the PINN reduced interpolation error by approximately $T_{N+1}$8 for $T_{N+1}$9 m. When its interpolated virtual signals replaced physical in-ROI error microphones in FxLMS, the steady-state ear-level noise reduction improved by about $$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$0, and the residual field around the ear regions was approximately $$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$1 lower.

A related systems use of interpolation appears in live free-viewpoint video streaming (Hu et al., 2021). There, view interpolation is not only an image-synthesis primitive but the mechanism that changes the architecture of the service. A synchronized 12-camera rig captures live video; a CNN-based midpoint interpolator, VINet, is recursively stacked so that $$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$2 stages produce $$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$3 views between adjacent cameras. The deployed system uses 4 stages, yielding 15 interpolated views between adjacent cameras and 177 total views across the 12-camera setup. These dense views are organized into multi-view cluster frames, encoded, segmented in time and viewpoint space, and streamed through HLS. The reported server-side average latencies were 12.85 ms for view interpolation, 1.27 ms for adaptive stitching, 5.62 ms for encoding, and 5.45 ms for scheduling, implying throughput of about 77 FPS. The architectural claim is that, once dense views are generated and packaged, server load becomes unrelated to the number of clients.

3. Frequency-domain interpolation for linear systems, identification, and passivity

In large-scale MIMO model reduction, system interpolation is explicitly frequency-domain and tangential (Jonas et al., 3 Jun 2025). For

$$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$4

the paper develops a left-tangential barycentric interpolant

$$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$5

that satisfies

$$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$6

for any admissible weight matrices $$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$7, while preserving the feedthrough $$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$8. The free weights are not chosen by direct minimization of $$x_k = \cos\left(\frac{2k-1}{2n}\pi\right), \quad k=1,2,\dots,n.$$9, but through a weighted $P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$0 surrogate,

$P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$1

The resulting quadratic problem has an explicit stationary condition

$P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$2

and the minimized surrogate value $P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$3 decreases monotonically with added interpolation rank: $P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$4 The framework is AAA-like in point addition, but uses low-rank tangential blocks, Gramian-based weight updates, and max-error, grid, or random point selection. On the 270-state ISS example, the max-error variant delivered $P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$5-error behavior close to balanced truncation and empirically produced stable reduced models on the tested stable systems.

Loewner rational interpolation provides a different realization-based paradigm for learning low-dimensional systems from frequency-response data (Drmač et al., 2019). In the SISO setting, the Loewner and shifted Loewner matrices are

$P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$6

leading to a descriptor interpolant with $P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$7 and $P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$8. Under relative complex Gaussian noise on the measurements, the paper proves that for each $P_N(x)=\sum_{i=0}^{N} c_i \bar{T}_i(x),$9,

$\phi_i(x)$0

with probability at least

$\phi_i(x)$1

provided a small-noise condition involving $\phi_i(x)$2, $\phi_i(x)$3, and $\phi_i(x)$4 holds. The practical message is that interpolation-point selection, left/right partitioning, and realization scaling directly affect robustness.

A related but identification-specific use of barycentric interpolation is the rapid frequency-domain scheme of (Jonas et al., 1 Oct 2025). Here the transfer function is unknown and each $\phi_i(x)$5 must be obtained by an expensive sinusoidal experiment. The identified interpolant is

$\phi_i(x)$6

with

$\phi_i(x)$7

so that $\phi_i(x)$8 when $\phi_i(x)$9. The paper exploits transient as well as steady-state data from each sinusoidal run and replaces the nonconvex output fit by the convex surrogate

$\{c_i\}$0

When $\{c_i\}$1 is positive definite, the unconstrained optimum is

$\{c_i\}$2

A Lyapunov-based LMI then enforces stability. On a 270-state ISS structural model, adaptive frequency selection outperformed gridded selection in $\{c_i\}$3 error.

Interpolation also underlies scalar positive-real synthesis. The modified Riccati method of (Cui et al., 2018) solves rational analytic interpolation with positivity and derivative constraints by reusing the covariance extension equation in the form

$\{c_i\}$4

The data enter through

$\{c_i\}$5

and the interpolant is reconstructed from

$\{c_i\}$6

For each spectral-zero polynomial $\{c_i\}$7 and feasible generalized Pick data, the solution is unique; moreover,

$\{c_i\}$8

Passivity-preserving interpolation is treated differently in (Benner et al., 2023). There the transfer function $\{c_i\}$9 is square and passive, and the key object is the para-Hermitian function

$T=[1\;\;0], \qquad s=[1\;\;1]^T.$0

Interpolation points and directions are obtained from a deflating subspace of the corresponding system pencil $T=[1\;\;0], \qquad s=[1\;\;1]^T.$1. The reduced model satisfies the bitangential conditions

$T=[1\;\;0], \qquad s=[1\;\;1]^T.$2

The paper then parameterizes a shifted family

$T=[1\;\;0], \qquad s=[1\;\;1]^T.$3

and shows that the corresponding reduced model has a stronger passivity certificate,

$T=[1\;\;0], \qquad s=[1\;\;1]^T.$4

which yields a normalized port-Hamiltonian realization with passivity-radius lower bound $T=[1\;\;0], \qquad s=[1\;\;1]^T.$5.

4. Structured nonlinear and parametric system interpolation

For polynomial and quadratic-bilinear systems, a single linear transfer function is inadequate, so interpolation is formulated on multivariate generalized transfer functions. In (Benner et al., 2019), polynomial parametric systems

$T=[1\;\;0], \qquad s=[1\;\;1]^T.$6

are associated with

$T=[1\;\;0], \qquad s=[1\;\;1]^T.$7

$T=[1\;\;0], \qquad s=[1\;\;1]^T.$8

and analogous $T=[1\;\;0], \qquad s=[1\;\;1]^T.$9. Projection spaces are then constructed so that the reduced model satisfies two-sided tangential Hermite interpolation, not only in the frequency variables $N=8$0 but also in the parameter $N=8$1. A Loewner-inspired realization is built from

$N=8$2

followed by SVD compression and effective projections $N=8$3. The numerical examples show that directly reducing the original cubic Chafee–Infante system is more accurate than first rewriting it in quadratic-bilinear form, with up to 3 orders of magnitude better accuracy; a parametric Chafee–Infante ROM of order 5 reproduces dynamics well across $N=8$4; and CUR approximation can reduce online cost for nonlinear evaluation.

A structurally parallel program is developed for quadratic-bilinear systems with internal differential structure (Benner et al., 2023). The paper uses generic operators

$N=8$5

to cover unstructured first-order QB systems, structured second-order systems, and time-delay QB systems. It defines structured symmetric subsystem transfer functions and structured generalized transfer functions, for example

$N=8$6

and derives exact interpolation conditions by requiring the projection bases to contain recursively defined subsystem responses such as $N=8$7, bilinear images $N=8$8, and quadratic images $N=8$9. Structure preservation is built in: $\sin(4x)+0.5\sin(8x)$0 On a quadratic time-delayed reaction-diffusion model, oversampled two-sided structured interpolation outperformed POD by about four orders of magnitude in time-domain accuracy and about six orders of magnitude on the first symmetric transfer function. On the Toda lattice molecular-dynamics example, preserving the internal block structure through split congruence transformations was important for stable reduced simulations.

5. Behavioral and direct data-driven trajectory interpolation

A distinct use of system interpolation appears in the behavioral LPV setting of (Verhoek et al., 15 Aug 2025). The systems considered are discrete-time LPV systems with shifted-affine dependence,

$\sin(4x)+0.5\sin(8x)$1

and the goal is to recover missing values in a partially specified finite trajectory without identifying an explicit parametric model. Given a data dictionary

$\sin(4x)+0.5\sin(8x)$2

the key representation theorem states that every admissible length-$\sin(4x)+0.5\sin(8x)$3 trajectory is characterized by a Hankel equation involving both $\sin(4x)+0.5\sin(8x)$4 and the lifted product signal $\sin(4x)+0.5\sin(8x)$5, under the generalized persistence of excitation condition

$\sin(4x)+0.5\sin(8x)$6

Interpolation is then formulated as exact missing-data completion for a given scheduling trajectory $\sin(4x)+0.5\sin(8x)$7 and a set $\sin(4x)+0.5\sin(8x)$8 of known entries. Existence is characterized by a rank/column-space compatibility condition, and uniqueness by

$\sin(4x)+0.5\sin(8x)$9

where $<1.1\%$0 spans the kernel of

$<1.1\%$1

Algorithmically, one solves a linear system for $<1.1\%$2,

$<1.1\%$3

and then reconstructs

$<1.1\%$4

The mass-spring-damper example illustrates the behavioral interpretation. With $<1.1\%$5, $<1.1\%$6, $<1.1\%$7, $<1.1\%$8, the paper identifies $<1.1\%$9, $T_{N+1}$00, $T_{N+1}$01, $T_{N+1}$02, and $T_{N+1}$03. For horizon $T_{N+1}$04, at least $T_{N+1}$05 data points were needed. With $T_{N+1}$06 known samples, Conditions 1–3 held and the interpolated trajectory matched the true trajectory up to numerical precision; with $T_{N+1}$07, existence held but uniqueness failed, yielding infinitely many admissible interpolants. This formulation treats simulation as a special case of interpolation and extends naturally to approximation when exact compatibility fails.

6. Interpolation in PDE regularity and solution families

In PDE analysis, system interpolation often means interpolation between function spaces rather than sample values. For the stationary 3D Navier–Stokes system in a bounded smooth domain,

$T_{N+1}$08

the paper (Degtyarev, 2024) uses the inequality

$T_{N+1}$09

to interpolate between $T_{N+1}$10 and $T_{N+1}$11. Taking $T_{N+1}$12, obtained from $T_{N+1}$13, the paper estimates the convection term $T_{N+1}$14 in $T_{N+1}$15 and derives

$T_{N+1}$16

Young’s inequality then yields

$T_{N+1}$17

with

$T_{N+1}$18

and the final a priori bound

$T_{N+1}$19

Here interpolation is the bridge between weak Sobolev control and Schauder regularity for a nonlinear system.

A different PDE meaning appears in the two-component Camassa–Holm system (Grunert et al., 2014). There, interpolation is between conservative and dissipative global weak solutions at wave breaking. In Lagrangian coordinates, the energy variable is split into $T_{N+1}$20 and an effective part $T_{N+1}$21. At each breaking time $T_{N+1}$22,

$T_{N+1}$23

with $T_{N+1}$24. Thus

$T_{N+1}$25

In Eulerian variables, the singular part of the energy measure is multiplied by $T_{N+1}$26 across breaking: $T_{N+1}$27 All $T_{N+1}$28-dissipative solutions solve the same 2CH system weakly; what changes is the fraction of concentrated energy retained after collision. This is a literal continuous interpolation of solution behavior at the system level.

7. Interpolation-generated dynamics, proof systems, and parameter-space cognition

In integrable-systems theory, interpolation can be the source from which the system itself is derived. The elliptic interpolation construction of (Yamada, 2017) starts from

$T_{N+1}$29

and asks for elliptic functions $T_{N+1}$30 such that

$T_{N+1}$31

Casorati determinants of $T_{N+1}$32 factor into residual spectral polynomials $T_{N+1}$33 and $T_{N+1}$34, whose zeros become dynamical variables. After gauge normalization, the contiguous relations $T_{N+1}$35 form a Lax pair, and compatibility yields the elliptic Garnier system. For $T_{N+1}$36, the construction reduces to the elliptic Painlevé equation. The multiple interpolation problem of (Doliwa, 2022) plays a closely related role for discrete integrable systems. There a determinant solution of

$T_{N+1}$37

produces $T_{N+1}$38-functions satisfying Hirota’s discrete KP equations and, after the interpolation-specific constraint is imposed, a non-autonomous multidimensional Toda system. In the $T_{N+1}$39 reduction, the resulting equations become the non-autonomous discrete-time Toda lattice.

Proof complexity introduces a logical meaning of effective system interpolation (Maxa, 6 Jan 2026). The paper studies the uniform effective disjunction property and the uniform effective interpolation property for $T_{N+1}$40 and stronger normal proof systems $T_{N+1}$41. If $T_{N+1}$42 has the uniform effective disjunction property, then every sufficiently strong $T_{N+1}$43 corresponding to a theory $T_{N+1}$44 in the language of $T_{N+1}$45 also has it. If $T_{N+1}$46 has the uniform effective interpolation property, then so does $T_{N+1}$47. A major consequence is that, under this hypothesis, every disjoint $T_{N+1}$48-pair has a separator in $T_{N+1}$49, and therefore

$T_{N+1}$50

The paper also proves that if $T_{N+1}$51 is an $T_{N+1}$52-pair with $T_{N+1}$53, then an exponential-time algorithm can, for each input $T_{N+1}$54, find some $T_{N+1}$55 such that $T_{N+1}$56. Here “interpolation” is not approximation but effective proof-theoretic extraction.

A final, very different use appears in checkpoint interpolation for LLMs (Yang et al., 29 Jan 2026). The paper defines a parameter-space interpolation between an Instruct checkpoint $T_{N+1}$57 and a Thinking checkpoint $T_{N+1}$58,

$T_{N+1}$59

and then makes the coefficient query-dependent,

$T_{N+1}$60

Its pilot study reports a convex, monotonic Pareto frontier between accuracy and token usage, with representation continuity supported by a correlation $T_{N+1}$61 between the first PCA component of the $T_{N+1}$62 representation and $T_{N+1}$63. The DAMI framework estimates $T_{N+1}$64 either by preference learning over accuracy/cost tradeoffs or by a zero-shot confidence discrepancy signal,

$T_{N+1}$65

On five mathematical reasoning benchmarks, the paper reports that DAMI-Pref improves accuracy over the Thinking model by 3.4 points with 29% fewer tokens, while DAMI-Conf improves accuracy by 2.5 points with 40% fewer tokens. This suggests a parameter-space notion of system interpolation in which the interpolated object is not a transfer function or trajectory but the model’s cognitive configuration itself.

Across these domains, system interpolation is less a single technique than a recurring structural idea: interpolation becomes “systemic” when it is governed by architecture, dynamics, admissibility, proof theory, or parameter geometry, and when its output modifies how the system as a whole behaves.