Physics-Aware Composite Kernels

• Physics-aware composite kernels are frameworks that integrate physical principles with flexible kernel design to encode domain-specific structure.
• They employ constructions like mixtures, products, hybrid models, and operator-induced regularization to balance mechanistic priors with data-driven adaptation.
• Applications in detector tracking, impact localization, and PDE-informed learning demonstrate improvements in accuracy, efficiency, and interpretability.

A physics-aware composite kernel is a kernel construction in which the similarity measure, covariance function, or induced reproducing-kernel Hilbert space (RKHS) is assembled so that physically meaningful structure is represented jointly with flexible data-driven variation. In recent work, the term spans several related usages: explicit mixtures or products of kernels chosen to reflect complementary physical effects, hybrid models that combine a mechanistic term with an RKHS correction, kernels induced by composite regularization involving differential operators, regime-switching Gaussian-process (GP) kernels that activate physics only where it is valid, and broader local transition kernels for compositional simulation (Oli et al., 2024, Donati et al., 9 Sep 2025, Doumèche et al., 2024, Pitchforth et al., 13 Jun 2025, Carlo et al., 20 Aug 2025, Hildebrandt et al., 7 May 2026). The unifying theme is that kernel design is treated not as a generic smoothness prior, but as a direct carrier of domain structure.

1. Kernel-theoretic scope

In the dominant statistical usage, a kernel is a symmetric positive semidefinite function $k:\mathcal X\times\mathcal X\to\mathbb R$ satisfying

$$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$

and it defines an RKHS of admissible functions. In this view, the kernel is simultaneously a similarity measure and a specification of the function space in which regression or classification is performed. A central practical consequence is closure: valid kernels remain valid under addition, multiplication, and linear transformations, which makes composite constructions mathematically straightforward (Noack et al., 2021).

This RKHS interpretation underlies multiple physics-aware settings. In SVM classification for charged-particle tracking, the kernelized decision rule is

$f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$

so the kernel directly determines the detector-level decision boundary used to reject low-$p_T$ charge clusters (Oli et al., 2024). In GP regression, the posterior mean and variance are functions of the kernel matrix and therefore inherit whatever physical structure has been encoded into the covariance (Xiao et al., 30 Jan 2025). In PDE-regularized learning, the physics penalty itself induces the RKHS norm, so the kernel is defined by the physical operator rather than selected ad hoc (Doumèche et al., 2024, Doumèche et al., 2024).

A broader probabilistic usage appears in radiation-matter simulation, where BRICKS defines a local stochastic interaction rule

$q_\phi(y\mid \theta)\approx p(y\mid \theta)$

for one-to-many particle transitions conditioned on a local material descriptor. Here “kernel” denotes a compositional Markov transition kernel rather than a positive semidefinite covariance kernel, but the role is analogous: it is the reusable local rule from which large-scale behavior is composed autoregressively (Hildebrandt et al., 7 May 2026).

2. Principal construction patterns

Across the literature, physics-aware composite kernels are not a single canonical formula but a family of constructions. Some are explicit sums or products of hand-designed kernels; some are hybrid mechanistic-plus-nonparametric models; others arise implicitly from regularizers or operator constraints.

Family	Representative construction	Representative sources
Mixed similarity kernels	$(1-r)\mathrm{SK}+r\mathrm{GK}$, $k_{comp}=k_{rbf} * k_{cos}$	(Oli et al., 2024, Xiao et al., 30 Jan 2025)
Model-plus-correction kernels	$$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$	(Donati et al., 9 Sep 2025)
Operator-induced kernels	RKHS norm $$\lambda_n\|f\|_{H^s}^2+\mu_n\|\mathscr D(f)\|_{L^2}^2$$ induces $K$	(Doumèche et al., 2024, Doumèche et al., 2024)
Regime-selective kernels	$$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$0	(Pitchforth et al., 13 Jun 2025)
Deep composite kernels	$$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$1, or $$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$2	(Jiang et al., 2022, Carlo et al., 20 Aug 2025)

The simplest explicit example is the mixed SVM kernel for silicon-tracker data,

$$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$3

which blends sigmoid and Gaussian behavior so that global and local structure are both available to the classifier (Oli et al., 2024). A closely related product form appears in impact localisation on composite aerostructures,

$$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$4

so that similarity is high only when two TDOA vectors are simultaneously close in magnitude and aligned in pattern or order (Xiao et al., 30 Jan 2025).

A different pattern is the grey-box composite model used for nonlinear system identification: $$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$5 The kernel no longer replaces the physical model; it spans an RKHS correction $$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$6 that absorbs unmodeled dynamics while the physical parameters $$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$7 are estimated jointly (Donati et al., 9 Sep 2025).

In PDE-informed regression, the composite character is in the regularization rather than a sum or product of named kernels. The norm

$$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$8

defines the RKHS geometry itself, yielding a kernel that is operator-adapted and domain-aware. One source explicitly notes that this is not a composite kernel in the usual $$\sum_i^N\sum_j^N c_i c_j\,k(\mathbf x_i,\mathbf x_j)\ge 0,$$9 or $f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$0 sense, but rather a single kernel induced by composite regularization (Doumèche et al., 2024, Doumèche et al., 2024).

Regime-selective GP kernels introduce a further pattern: the physical kernel is turned on only when a switching variable indicates that the corresponding mechanism is active. With a sigmoid gate,

$f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$1

the covariance interpolates smoothly between a physics-derived component and a fallback data kernel, so the relative reliance on physics varies through operating regimes (Pitchforth et al., 13 Jun 2025).

3. Sources of physical structure

The defining property of these kernels is not simply compositionality, but the origin of their components in measurable physical structure.

In collider tracking, the SVM features are detector-native rather than abstract embeddings. Each charge cluster is represented by 14 features: 13 from the $f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$2-profile of deposited charge in the pixel rows and one feature $f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$3, the distance in the $f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$4-axis between the impact point and the center of the flat module. The labels are also physically defined: high-$f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$5 for $f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$6, low-$f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$7 otherwise. The kernel is therefore tailored to charge-sharing geometry, track incidence, and curvature, not generic classification geometry (Oli et al., 2024).

In impact localisation, the physical prior comes from flexural-wave propagation in laminated composites. For low-frequency flexural waves,

$f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$8

and the corresponding TDOA values scale approximately with the same factor, which implies that changes in frequency or temperature may scale TDOA magnitudes without changing arrival-time order. This motivates a cosine-similarity factor that encodes order or angular similarity, alongside an RBF factor for magnitude-based similarity. The same physical argument also underlies the preference for sample standardisation,

$f(x)=\sum_{s\in SV}\alpha_s y_s K(x_i,u)+b,$9

because it preserves within-sample order structure and compatibility with the cosine kernel (Xiao et al., 30 Jan 2025).

In structural dynamics, the relevant physical structure is often regime-limited. A lift-force model may be valid only for high-speed side-on winds, or oscillatory aircraft-wing dynamics may matter only during rudder-induced manoeuvres. The composite change-point kernel therefore uses a separate switching input $p_T$0 so that the gate depends on interpretable conditions such as wind direction, wind speed, temperature, humidity, turbulence, or pilot control input. Physics is encoded not merely by a structured covariance, but by a physically meaningful activation condition (Pitchforth et al., 13 Jun 2025).

In steering-vector super-resolution for augmented listening, the field is decomposed into direct and scattering components,

$p_T$1

and the GP kernel mirrors that decomposition: $p_T$2 The direct-path term is built from the free-field steering vector, while the scattering term is expressed through spherical harmonics coefficients predicted by a neural field. Frequency smoothness, propagation delay, array geometry, and HRTF-like scattering all appear explicitly in the covariance (Carlo et al., 20 Aug 2025).

At a more abstract level, physics can be encoded through operators and symmetries. PDE-informed kernel learning constructs $p_T$3 from the differential operator $p_T$4, so modes that violate the PDE are penalized spectrally (Doumèche et al., 2024, Doumèche et al., 2024). Domain-aware GP design uses linear operators to enforce symmetry or periodicity exactly in the RKHS, and additive or anisotropic kernels to reflect separability or direction-dependent length scales (Noack et al., 2021).

4. Balancing physical priors and data adaptation

A recurrent issue is the relative balance between mechanistic structure and data-driven flexibility. Several papers make this balance explicit in the objective or kernel parameters.

In nonlinear system identification, the joint estimator

$p_T$5

uses $p_T$6 to regulate how much freedom the RKHS correction has relative to the physical model. Small $p_T$7 gives the kernel more freedom; large $p_T$8 forces the solution closer to the nominal physical structure. The same paper explicitly contrasts this with two-step discrepancy modeling, which first estimates $p_T$9 with $q_\phi(y\mid \theta)\approx p(y\mid \theta)$0 and then fits residuals, thereby producing biased physical parameters (Donati et al., 9 Sep 2025).

In regime-switching GPs, the balance is controlled by sigmoid hyperparameters $q_\phi(y\mid \theta)\approx p(y\mid \theta)$1 and $q_\phi(y\mid \theta)\approx p(y\mid \theta)$2, which determine transition sharpness and location. These can be specified by the user when the regime threshold is known, or learned by maximizing the GP marginal likelihood when the threshold is uncertain. The same framework also allows heteroscedastic noise to vary with the physical regime via a second GP on the noise variance, so both the latent function and the uncertainty become condition-dependent (Pitchforth et al., 13 Jun 2025).

In robust impact localisation, no single kernel is forced to dominate. Instead, Bayesian model averaging combines predictions from multiple kernels through weights based on marginal likelihood and predictive uncertainty,

$q_\phi(y\mid \theta)\approx p(y\mid \theta)$3

with final weight

$q_\phi(y\mid \theta)\approx p(y\mid \theta)$4

This gives a dynamic fusion mechanism in which the RBF, cosine, or composite input kernel can dominate depending on condition and uncertainty (Xiao et al., 30 Jan 2025).

Other works expose the balance through architecture or parameterization rather than a single scalar. The Implicit Composite Kernel splits the input into a high-dimensional source processed by a neural network and a low-dimensional source processed by an explicit kernel approximation, yielding an approximate GP prior with multiplicative covariance $q_\phi(y\mid \theta)\approx p(y\mid \theta)$5. The deep part learns content; the explicit kernel supplies known structure such as periodicity or smoothness (Jiang et al., 2022). In detector filtering, the mixing ratio $q_\phi(y\mid \theta)\approx p(y\mid \theta)$6 determines whether the SVM behaves like a pure sigmoid kernel, a pure Gaussian kernel, or an interpolation of the two (Oli et al., 2024).

5. Estimation algorithms and computational realization

Physics-aware composite kernels often impose more structure than standard kernels, so computational strategy becomes part of the design.

For partially observed nonlinear state-space systems, the composite kernel identification framework first performs forward filtering with an Unscented Kalman Filter and then backward smoothing with an Unscented Rauch–Tung–Striebel smoother. After smoothing, the dynamic identification problem is recast as a static one with an effective input $q_\phi(y\mid \theta)\approx p(y\mid \theta)$7, and the same parametric-plus-kernel estimator is applied. This avoids direct multi-step nonlinear optimization over hidden states and model parameters simultaneously (Donati et al., 9 Sep 2025).

For PDE-regularized kernel regression, the exact kernel is generally intractable, so physics-informed kernel learning approximates it in a truncated Fourier basis on a periodic extension of the domain. The resulting estimator has a closed-form representer expression involving a matrix $q_\phi(y\mid \theta)\approx p(y\mid \theta)$8 that combines Sobolev and PDE-penalty terms in Fourier space, so the physical prior acts as a structured spectral filter (Doumèche et al., 2024).

KP-PINNs adopt a different route. They replace the standard PINN $q_\phi(y\mid \theta)\approx p(y\mid \theta)$9 residual loss by an RKHS norm associated with a tensor-product Matérn kernel, leading to quadratic forms such as

$(1-r)\mathrm{SK}+r\mathrm{GK}$0

Kernel Packet acceleration exploits sparse banded structure to compute the inverse efficiently. The paper reports $(1-r)\mathrm{SK}+r\mathrm{GK}$1 complexity for computing the relevant matrices, $(1-r)\mathrm{SK}+r\mathrm{GK}$2 storage, and $(1-r)\mathrm{SK}+r\mathrm{GK}$3 total inversion via LU on the banded structure (Yang et al., 10 Jun 2025).

Deep composite kernels also require approximation machinery. The Implicit Composite Kernel uses Nyström approximation,

$(1-r)\mathrm{SK}+r\mathrm{GK}$4

or random Fourier features to map an explicit kernel into a latent inner-product representation that can be trained end-to-end with a neural network. Its uncertainty approximation follows a sample-then-optimize ensemble interpretation (Jiang et al., 2022). In GP steering-vector regression, a neural field parameterizes spherical-harmonic coefficients inside the scattering kernel, while inference proceeds through the GP marginal likelihood (Carlo et al., 20 Aug 2025).

An especially distinctive realization appears in high-energy physics hardware co-design. The mixed-kernel SVM is mapped to a mixed-kernel heterojunction transistor built from MoS$(1-r)\mathrm{SK}+r\mathrm{GK}$5 and carbon nanotubes. The paper describes this 2D van der Waals heterojunction device as capable of Gaussian, sigmoid, and mixed-kernel-like behavior in a single analog component, with estimated mean power for kernel operation of approximately $(1-r)\mathrm{SK}+r\mathrm{GK}$6 and roughly two orders of magnitude greater power efficiency than traditional analog CMOS circuits for similar kernel generation tasks. The broader analog mapping uses resistive-memory crossbar arrays for dot products, the mixed-kernel transistor for nonlinearity, and a support-vector weighted sum for the final decision (Oli et al., 2024).

6. Representative applications and reported behavior

The diversity of application areas is one of the clearest indicators that physics-aware composite kernels are a methodological family rather than a domain-specific trick.

In detector-level data reduction for silicon tracking, the mixed-kernel SVM is trained on simulated smart pixel sensor charge deposits due to pions from a $(1-r)\mathrm{SK}+r\mathrm{GK}$7 pixel region of a $(1-r)\mathrm{SK}+r\mathrm{GK}$8 flat silicon sensor. It achieves $(1-r)\mathrm{SK}+r\mathrm{GK}$9 signal efficiency, with background rejection of $k_{comp}=k_{rbf} * k_{cos}$0 on track data, $k_{comp}=k_{rbf} * k_{cos}$1 on untracked data, and $k_{comp}=k_{rbf} * k_{cos}$2 on single-pixel-hit data. Using the stated category mixture, the paper estimates an overall background rejection of $k_{comp}=k_{rbf} * k_{cos}$3 while maintaining $k_{comp}=k_{rbf} * k_{cos}$4 signal efficiency (Oli et al., 2024).

In nonlinear system identification, the simultaneous physics-plus-kernel estimator improves both parameter recovery and predictive performance. On a synthetic regression example, ordinary least squares gives RMSE $k_{comp}=k_{rbf} * k_{cos}$5, two-step discrepancy modeling gives RMSE $k_{comp}=k_{rbf} * k_{cos}$6, and the proposed simultaneous method gives RMSE $k_{comp}=k_{rbf} * k_{cos}$7. On the cascade tank system benchmark, adding the kernel to the nominal physics model changes prediction RMSE from about $k_{comp}=k_{rbf} * k_{cos}$8 on training and $k_{comp}=k_{rbf} * k_{cos}$9 on validation; in simulation it changes RMSE from about $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$0 on training and $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$1 on validation, with fit improving from roughly $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$2 to over $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$3 (Donati et al., 9 Sep 2025).

In robust impact localisation on composite aerostructures under temperature variation with six sensors, the multiplicative composite kernel outperforms either constituent kernel in mean error: COMP gives mean localisation error $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$4 mm and maximum $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$5 mm, COS gives mean $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$6 mm and maximum $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$7 mm, RBF gives mean $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$8 mm and maximum $$\hat y(x)=f(x,\theta^\star)+\sum_{j=1}^T \omega_j^\star \kappa(x,x_j)$$9 mm, and Bayesian model averaging gives mean $$\lambda_n\|f\|_{H^s}^2+\mu_n\|\mathscr D(f)\|_{L^2}^2$$0 mm and maximum $$\lambda_n\|f\|_{H^s}^2+\mu_n\|\mathscr D(f)\|_{L^2}^2$$1 mm. The reported interpretation is that cosine similarity is robust when order information dominates, RBF is stronger near singularity, and the product kernel performs best overall because it captures both (Xiao et al., 30 Jan 2025).

In structural dynamics, physically informed change-point kernels improve NMSE and MSLL over a plain squared exponential GP on the Tamar bridge case study and reconstruct higher-frequency manoeuvre content in aircraft wing strain by switching in a physics-derived SDOF kernel during high rudder angles. The learned switching function identifies an interpretable activation window around roughly $$\lambda_n\|f\|_{H^s}^2+\mu_n\|\mathscr D(f)\|_{L^2}^2$$2–$$\lambda_n\|f\|_{H^s}^2+\mu_n\|\mathscr D(f)\|_{L^2}^2$$3 of rudder angle (Pitchforth et al., 13 Jun 2025).

In PDE-informed learning, the kernel formulation can materially outperform PINNs. For the 1D wave equation, one reported comparison gives PINN error around $$\lambda_n\|f\|_{H^s}^2+\mu_n\|\mathscr D(f)\|_{L^2}^2$$4, PIKL error around $$\lambda_n\|f\|_{H^s}^2+\mu_n\|\mathscr D(f)\|_{L^2}^2$$5, and a training-time speedup of about $$\lambda_n\|f\|_{H^s}^2+\mu_n\|\mathscr D(f)\|_{L^2}^2$$6. The same work emphasizes that PIKL can be particularly advantageous with noisy boundary conditions, where classical PDE solvers may propagate noise (Doumèche et al., 2024).

In steering-vector super-resolution for augmented listening, GP-Steerer uses a composite kernel over frequency, microphone position, and source direction to recover continuous steering-vector fields from sparse measurements. The paper reports that it outperforms all listed baselines, especially at high frequencies and sparse sampling, and attains oracle-level or near-oracle performance in downstream tasks with fewer than ten times fewer measurements (Carlo et al., 20 Aug 2025).

7. Limitations, design tensions, and open directions

The recent literature consistently treats the balance between physical prior and flexible correction as a design problem rather than a solved principle. One paper states this directly: over-reliance on physical knowledge can be detrimental when the physics-based component is inaccurate, while underutilisation wastes interpretability and increases data demand (Pitchforth et al., 13 Jun 2025). The same tension appears in regularization parameters such as $$\lambda_n\|f\|_{H^s}^2+\mu_n\|\mathscr D(f)\|_{L^2}^2$$7 in joint system identification, where too small a value permits overfitting by the kernel and too large a value enforces excessive trust in the nominal model (Donati et al., 9 Sep 2025).

Structural richness also increases optimization burden. In the Tamar bridge study, the physics-informed change-point kernel increases the number of hyperparameters from 3 for the squared-exponential baseline to 11, with computation time rising by about a factor of 1.7 (Pitchforth et al., 13 Jun 2025). In the tracking SVM, increasing the number of support vectors can improve classification performance but raises area and power cost in hardware (Oli et al., 2024). In ICK, performance degrades when the number of multiplied sources exceeds three, which the paper attributes to vanishing gradients, and the Nyström-based predictive variance is not guaranteed to remain positive semidefinite in all cases (Jiang et al., 2022).

A further limitation concerns kernel misspecification. Regime-selective kernels depend on choosing a meaningful switching variable and an appropriate physical kernel; if either choice is poor, the benefits diminish (Pitchforth et al., 13 Jun 2025). Standard stationary kernels may themselves be a source of misspecification, since they impose translation invariance and, in the squared-exponential case, infinite differentiability regardless of whether the underlying physics supports it. Domain-aware GP design therefore emphasizes additive, anisotropic, symmetry-enforcing, periodic, and non-stationary constructions when stationarity is unrealistic (Noack et al., 2021).

A plausible implication is that future work will continue to move away from generic smooth kernels toward structured kernel geometries whose components, gates, operators, and hyperparameters are physically interpretable. The surveyed literature already shows several mature variants of that idea: detector-aware mixtures, order-aware similarity products, physics-plus-RKHS discrepancy models, operator-defined kernels, regime-selective covariances, and deep kernels in which learned representations parameterize physically structured covariance factors (Oli et al., 2024, Xiao et al., 30 Jan 2025, Donati et al., 9 Sep 2025, Doumèche et al., 2024, Pitchforth et al., 13 Jun 2025, Carlo et al., 20 Aug 2025).