--- title: Physics-Informed Kernels (PIKs) url: https://www.emergentmind.com/topics/physics-informed-kernels-piks type: topic --- # Physics-Informed Kernels (PIKs) Physics-informed kernels (PIKs) are kernelized constructions in which physical structure is embedded directly into the representation or evolution law used by a learning algorithm. Across the recent literature, the term denotes several related but non-identical objects: in physics-informed neural solvers it can mean a fixed or trainable feature map that lifts coordinates into Fourier, wavelet, Chebyshev, spline, or radial-basis spaces; in kernel learning it can mean the reproducing kernel induced by a PDE-regularized variational problem; in Gaussian-process models it can mean a covariance function modified by orthogonality, discrepancy, or change-point constraints; and in generative sampling it can mean the flow field that transports a simple distribution to a target one while preserving probability weight [2512.12001] [2510.26678] [2409.13786] [2506.11625]. A common source of confusion is therefore terminological: “kernel” does not refer to a single formalism, but to several mathematically distinct mechanisms for incorporating physics. ## 1. Definitions and conceptual scope In the PINN and PIKAN literature, a physics-informed kernel is a feature map applied to an input coordinate before the coordinate is processed by the network. In this sense, the kernel lifts each scalar input \(x\) into a richer function space, such as Fourier or wavelet bases, so that differential operators act in known ways on the embedded features. In PIKANs the same terminology is used for the pre-activated univariate maps \(\phi_{ij}(x_i)\) in the Kolmogorov–Arnold decomposition; when those maps are chosen or augmented with physics-informed spectral kernels, the resulting model is called a Physics-Informed Kernel-Arnold Network [2512.12001]. In the kernel-method literature, the term refers instead to the RKHS kernel associated with a PDE-regularized estimator. There the starting point is an empirical risk augmented by Sobolev and differential-operator penalties, and the corresponding minimizer is shown to lie in the RKHS of a uniquely defined kernel. The kernel is therefore not merely an architectural prior but the Green’s-function or weak-solution object associated with the regularized variational problem [2402.07514] [2409.13786]. In generative sampling, a PIK is the infinitesimal generator \(\dot\phi_t(\phi)\) of a renormalisation-group flow. That object is defined so as to preserve a prescribed path of probability measures between a base distribution and a target distribution. In the sign-problem literature, the same flow-field view is used to construct mappings to sign-problem–free manifolds with a probability-weight preserving property [2510.26678] [2603.03159]. Other works use “physics-informed kernel” in still different but structurally related ways. PIKFNNs employ PDE-derived kernel functions as activation functions satisfying the governing equation in the interior [2306.02606]. KAPI-ELM uses adaptive Gaussian RBF kernels whose distributional hyperparameters are optimized by Bayesian optimization [2507.10241]. Physically-informed change-point kernels modulate the contribution of a physics-based covariance term through logistic gating [2506.11625]. Orthogonal discrepancy kernels subtract the projection of a Gaussian-process discrepancy onto a physics subspace so that the GP does not explain structure already assigned to the mechanistic model [2606.21199]. In graph-based detector filtering, interaction kernels \(J_{ij}\) encode propagation constraints from geometry and kinematics inside an Ising-type energy functional [2603.24075]. This suggests that PIKs are best understood as an umbrella family of constructions rather than a single model class. ## 2. Spectral feature embeddings and physics-informed Kolmogorov–Arnold networks A central neural-solver interpretation of PIKs arises in PIKANs. By the Kolmogorov–Arnold representation theorem, any continuous \(f:\mathbb R^n\to\mathbb R\) can be written as \[ f(x_1,\dots,x_n)=\sum_{j=1}^{m}\psi_j\Bigl(\sum_{i=1}^n \phi_{ij}(x_i)\Bigr), \] with univariate functions \(\phi_{ij}\) and \(\psi_j\). In a PIKAN, the univariate transforms are small neural sub-units, and the “kernel” language refers to the choice or augmentation of these input-side transforms by physics-informed basis functions [2512.12001]. The HWF-PIKAN architecture makes this explicit. Each scalar input \(u\in[0,1]\) is first normalized and then embedded by Fourier and wavelet features. The Fourier map is \[ \gamma_F(u)=\bigl[\sin(2\pi k\,u),\;\cos(2\pi k\,u)\bigr]_{k=1}^{M_{\rm Fourier}}, \] which corresponds to the real and imaginary parts of \(K_{\rm Fourier}(u,\omega_k)=e^{i\omega_k u}\) with \(\omega_k=2\pi k\). The wavelet map uses the Ricker or Mexican-hat construction \[ \psi_{j,k}(u)=\Bigl(1-\bigl(\tfrac{u-c_{j,k}}{s_j}\bigr)^2\Bigr)\exp\!\Bigl[-\tfrac12\bigl(\tfrac{u-c_{j,k}}{s_j}\bigr)^2\Bigr], \] with uniformly spaced centers \(c_{j,k}\) at scale \(j\) and \(s_j=O(2^{-j})\). The hybrid embedding concatenates the two: \[ \gamma_{\rm HWF}(u)=\bigl[\gamma_F(u),\{\psi_{j,k}(u)\}_{j,k}\bigr]. \] In multiple dimensions the same map is applied coordinate-wise and then flattened [2512.12001]. The forward pass of HWF-PIKAN is organized as normalization, construction of Fourier and wavelet embeddings, concatenation into \(\gamma_{\rm HWF}\), layer normalization, processing by the KAN/B-spline core implementing the successive univariate maps \(\phi^k\), and output of \(\hat u_\theta(\mathbf z)\). The loss has the standard physics-informed decomposition \[ \mathcal L=\lambda_{\rm PDE}\mathcal L_{\rm PDE}+\lambda_{\rm IC}\mathcal L_{\rm IC}+\lambda_{\rm BC}\mathcal L_{\rm BC}, \] and for the collisionless Boltzmann equation in 1D spatio-velocity form the residual is \[ \mathcal R_{\rm PDE}(x,v,t)=f_t(x,v,t)+v\,f_x(x,v,t). \] Training proceeds by an IC warm-up minimizing \(\mathcal L_{\rm IC}\) with Adam, then Adam on the full loss, optionally followed by L-BFGS on the combined collocation sets. Collocation points are drawn uniformly or via Latin-Hypercube, with \(N_r\) typically \(10^4\)–\(3\times 10^5\). The reported typical hyperparameters for 1D and 2D advection are \(M_{\rm Fourier}=32\), \(J_{\rm wavelet}=3\), \(Q=16\), \(p=3\), and \(n_{\rm control}=25\) [2512.12001]. The empirical motivation is mitigation of spectral bias. In the HWF-PIKAN study, Fourier modes capture smooth global waves, while wavelets capture localized jumps and filaments. On the Sod shock-tube over CBE, the reported loss at 100 epochs is \(1.23\) for PINN, \(0.933\) for PIKAN, \(1.68\) for F-PIKAN, \(0.805\) for W-PIKAN, and \(0.707\) for HWF-PIKAN; at 10,000 epochs the corresponding values are \(0.00320\), \(0.00180\), \(0.00140\), \(0.00120\), and \(0.00090\). The same paper reports that in one- and two-dimensional advection benchmarks HWF-PIKAN achieves lower MSE in fewer epochs and fewer collocation points, and that in phase-space CBE problems it reduces loss by up to an order of magnitude compared to vanilla PINN and by 20–40% compared to single-type embeddings [2512.12001]. The NTK analysis of cPIKANs provides a complementary theoretical account. In supervised form, \[ K_{ij}(\tau)=\bigl\langle \nabla_\theta f(x_i;\theta(\tau)),\,\nabla_\theta f(x_j;\theta(\tau))\bigr\rangle, \] and in the physics-informed setting the full kernel becomes the block matrix \[ \mathcal K(\tau)= \begin{pmatrix} K_{uu} & K_{ur}\\ K_{ru} & K_{rr} \end{pmatrix}, \] coupling prediction and residual dynamics. For Helmholtz, diffusion, Allen–Cahn, and Euler–Bernoulli benchmarks, cPIKANs show broader and better-conditioned spectra than standard PINNs. One reported textual figure gives \(K_{rr}\) eigenvalue decay \(\lambda_\ell\propto \ell^{-0.3}\) for cPIKAN versus \(\ell^{-1.2}\) for PINN on Helmholtz. In Allen–Cahn, time-domain decomposition reduces \(\lambda_{\max}/\lambda_{\min}\) from \(\sim 10^5\) on a single domain to \(\sim 10^3\) on four subdomains, cutting training time and error by over 95%. Hybrid Adam-to-L-BFGS optimization is reported to yield the largest effective rank and the smallest condition number of the NTK [2506.07958]. ## 3. RKHS, Fourier, and deep-kernel formulations A second major meaning of PIKs treats physics-informed learning as a kernel method. The starting point is a regression or PDE-solving objective of the form \[ R_n(f)=\frac1n\sum_{i=1}^n |f(x_i)-Y_i|^2+\lambda_n\|f\|_{H^s(\Omega)}^2+\mu_n\|\mathcal D f\|_{L^2(\Omega)}^2, \] where \(\mathcal D\) is a known linear differential operator. The minimizer is shown to lie in the RKHS of a uniquely defined kernel \(K\), and one may write \[ \hat f_n(x)=[K(x,X_1),\dots,K(x,X_n)]\,(K+nI_n)^{-1}Y. \] Equivalently, the kernel may be described as the Green’s function associated with \(\lambda_n(1-\Delta)^s+\mu_n\,\mathcal D^*\mathcal D\) or, in the weak formulation, as the unique solution of the PDE induced by the regularized bilinear form [2402.07514] [2409.13786]. The Fourier construction used in physics-informed kernel learning makes this explicit. On a torus extension of the domain, one truncates the periodic Fourier basis \(\phi_k(x)\) to \(\|k\|_\infty\le m\) and defines a matrix \(M_m\) whose entries combine Sobolev damping and the polynomial symbol \(P(k)\) of \(\mathcal D\). The resulting estimator is \[ \hat f_{\rm PIKL}(x)=\Phi_m(x)^*\,(\Phi^*\Phi+nM_m)^{-1}\Phi^*Y. \] This produces a closed-form estimator with storage \(O((2m+1)^{2d})\) and computation \(O(n(2m+1)^d+(2m+1)^{3d})\), and the reported approximation error satisfies \(\|K-K_m\|_\infty=O(m^{-2(s-d/2)})\) for \(s>d/2\) [2409.13786]. Theoretical guarantees are stated in terms of the effective dimension. Without exploiting PDE exactness, one recovers the Sobolev minimax rate \(n^{-2s/(2s+d)}\). When the physical prior is exact or nearly exact, substantially faster convergence is possible. One result gives the nearly parametric rate \(O(n^{-1}\log^3 n)\) for perfect PDE modeling in PIKL, and the companion RKHS analysis states that the ideal exact-model case can achieve the parametric rate \(n^{-1}\) up to logarithmic factors [2409.13786] [2402.07514]. In a one-dimensional example with \(\mathcal D=\frac d{dx}\), the integral-operator eigenvalues satisfy \(a_m\asymp ((\lambda_n+\mu_n)m^2)^{-1}\), and the corresponding bound becomes \[ \E\|\hat f_n-f^\star\|^2 \lesssim \|\mathcal D f^\star\|\,n^{-2/3}\log^3 n + (\|f^\star\|_{H^1}^2+\sigma^2)\,n^{-1}\log^3 n, \] with the \(n^{-1}\) term dominating when \(\mathcal D f^\star=0\) [2402.07514]. The numerical record reported for PIKL is unusually strong in low-dimensional linear-PDE settings. For a harmonic oscillator hybrid-modeling problem, PIKL with \(m=300\) and \(s=2\) exhibits an experimental \(n^{-1.1}\) rate, matching OLS on the true two-dimensional solution manifold. For the 1D convection equation with periodic boundary conditions, PIKL with \(n=100\) and \(m=20\) achieves \(\approx 10^{-7-8}\), whereas vanilla PINN gives \(\approx O(10^{-1})\) and curriculum PINN \(\approx O(10^{-2})\). For the 1D wave equation with Dirichlet boundaries, PIKL gives \(\approx 8.7\times 10^{-4}\) in \(6\) s, while the cited vanilla PINN has error \(\approx 0.45\) in \(7\) min and an NTK-optimized PINN \(\approx 1.7\times 10^{-3}\). In noisy-boundary wave problems with variance \(10^{-2}\), PIKL attains \(L^2\)-relative error \(\approx 1.87\times 10^{-2}\), while Euler, RK4, and Crank–Nicolson degrade to \(O(10^{-1-2})\) [2409.13786]. Deep-kernel variants extend the same theme to latent-source differential equations. In Physics Informed Deep Kernel Learning, one places a GP prior \(f\sim GP(0,k_\theta)\), uses posterior samples of \(f\) as surrogates for solutions of \(\psi[f](x)=g(x)\), and places a second GP prior on the induced latent source \(g\). After marginalization, the objective becomes a collapsed ELBO, \[ \mathcal L(\theta,\tau,\phi)=\log N(y\mid 0,K_{XX}+\tau^{-1}I)+\gamma\,\mathbb E_{X_s,\epsilon}\!\left[\log N\bigl(\psi[\tilde f](X_s;\epsilon)\mid 0,K_{ss}^\kappa\bigr)\right]. \] For linear operators one may also view the physics component through an operator-applied kernel \(k_L(x,x')=\mathcal L_x\mathcal L_{x'}k(x,x')\), although the surrogate-sampling formulation is introduced precisely to avoid analytic intractability for nonlinear \(\psi\) or deep kernels. The reported experiments show 50–80% reductions in extrapolation RMSE versus SKL, DKL, and latent force models on synthetic ODE and PDE tasks, along with improved uncertainty quantification on several real datasets [2006.04976]. ## 4. Renormalisation-group and flow-based PIKs In the generative-sampling literature, a PIK is not a covariance or feature map but the flow field \(\dot\phi_t(\phi)\) that reparametrizes a target distribution. One begins with a one-parameter family of normalized weights \(p_t(\phi)=\exp[-S_t(\phi)]\), and the kernel is defined implicitly by a continuity or Wegner equation. At the level of the action, the defining relation is \[ \frac{d S_t(\phi)}{dt}+\dot\phi_{t,i}(\phi)\,\partial_{\phi_i}S_t(\phi)=\partial_{\phi_i}\dot\phi_{t,i}(\phi). \] The global flow map is \[ \phi_t=\phi_0+\int_0^t ds\,\dot\phi_s(\phi_s), \] and the central property is exact preservation of weight along the transported manifold [2510.26678] [2603.03159]. The preservation law can be written locally as \[ e^{-S_t(\phi_t(\phi_0))}\,\det\!\left[\frac{\partial \phi_t}{\partial \phi_0}\right]=e^{-S_0(\phi_0)}, \] and globally as \[ \int_{\mathcal M} d\mu(\phi)\,p_1(\phi)\,O(\phi)=\int_{\mathcal M_0} d\mu_0(\phi_0)\,p_0(\phi_0)\,O(\phi_1(\phi_0)). \] The sign-problem paper emphasizes that the statistical weight of any flowing submanifold is exactly preserved, with \(dP_t(\mathcal T_t)/dt=0\). Under the stated assumptions—well-defined Wegner flow and no singularities crossed—this implies that no sign problem or overlap problem is introduced by the mapping, because oscillations in intermediate complex weights are exactly compensated by the Jacobian [2603.03159]. Algorithmically, this converts generative modeling into a sequence of linear layerwise solves. One chooses an analytic path \(S_t\), discretizes \(t\), and for each layer posits an ansatz \[ \dot\phi_t(\phi)=\sum_j k_{j,t}K_j(\phi), \] which turns the kernel equation into a linear system \(A_t k_t=b_t\). Residual-based corrections can then be obtained by solving another linear problem for \(\Delta\dot\phi_t\). The resulting depth-\(N\) network propagates samples by \[ \phi_{i+1}=\phi_i+\Delta t\sum_j k_{j,i}K_j(\phi_i), \] with optional higher-order ODE integrators and, if desired, a final Metropolis–Hastings step whose acceptance probability is reported to be very close to unity when the PDE solves are accurate [2510.26678]. The cited proof-of-principle benchmark is a zero-dimensional \(\phi^4\) theory flowing from a Gaussian base with \((m_0^2=1,\lambda_0=0)\) to a bistable target with \((m_1^2=-2,\lambda_1=0.5)\). With 51 layers and 25 sine-modes per layer, the mean-squared-error of the kernel equation falls below \(10^{-4}\), the effective sample size is \(\gtrsim 1\), the generated histogram matches \(10^6\) HMC reference samples almost exactly, and the runtime is reported to be \(\approx 20\times\) faster than HMC for the same number of independent samples [2510.26678]. The sign-problem extension applies the same framework to zero-dimensional field theories with complex couplings and to the real-time quantum-mechanical harmonic oscillator. In the oscillator example, the Wegner flow is linear, \[ \dot\phi_t(\phi)=-\tfrac12 M_t^{-1}\Bigl(\frac{dM_t}{dt}\Bigr)\phi, \] and \(10^5\) Euclidean paths sampled at \(t=0\) are transported to \(t=1\), yielding real-time two-point functions in perfect agreement with the analytic continuum. The comparison made in the paper is explicitly against Lefschetz-thimble methods, complex Langevin, and dual-variable rewritings: PIKs produce a single real-weight contour or “PIKfold,” while the tested models are reported to match exact or high-precision numerics where LT or CLE fail [2603.03159]. ## 5. Activation, adaptive-basis, and discretization kernels in PDE solvers A different line of work moves physical structure into the basis or activation itself. In Physics-Informed Kernel Function Neural Networks, one considers a linear PDE \[ \mathcal L u(x)=f(x), \qquad \mathcal B u(x)=g(x), \] and defines a physics-informed kernel function \(\phi(x;s)\) so that \[ \mathcal L_x \phi(x;s)=0,\qquad x\in \Omega\setminus\{s\}. \] The approximate solution is then \[ u_N(x)=b+\sum_{j=1}^N w_j\,\phi(x;s_j), \] which satisfies the homogeneous PDE interior by construction. Training therefore uses only boundary or initial data, for example \[ {\rm Loss}=\frac1{N_b}\sum_{i=1}^{N_b}\bigl(u_N(x_i)-g(x_i)\bigr)^2 \] in the Dirichlet case. The reported kernels include \(\phi(r)=\ln r\) for 2D Laplace and \(\phi(r)=J_0(kr)\) for 2D Helmholtz, together with constructions for nonhomogeneous and transient problems [2306.02606]. The benchmark suite for PIKFNN covers nine examples. The high-wavenumber 2D Helmholtz problem on \(\Omega=[-1,1]^2\) with \(k=50\sim 150\) uses \(N_b=80\) and \(J_0(k\|x-s_j\|)\), reaching \(L_2\)-error \(\approx 1.3\times 10^{-3}\). An infinite-domain Laplace problem on \(\mathbb R^2\setminus D\) with \(N_b=1600\) and \(\ln\|x-s_j\|\) reaches \(L_2\)-error \(\approx 8\times 10^{-12}\) in \(1.5\) s. Further examples include 3D nonhomogeneous Helmholtz in a rabbit geometry, a 3D transient heat problem on a torus with max relative error \(\approx 1.6\times 10^{-4}\), a 4D Laplace problem with \(R^2=1.0000\), inverse electromyography in pregnant sheep, and thin elastic plates with stress errors at a point \(\sim 10^{-9}\) [2306.02606]. KAPI-ELM uses another kernelized construction. The solution is expanded in Gaussian RBFs, \[ \hat u(x)=\sum_{i=1}^{N^*} c_i\,G_i(x;\alpha_i^*,\sigma_i),\qquad G_i(x;\alpha_i^*,\sigma_i)=\exp\!\Bigl(-\frac{\|x-\alpha_i^*\|^2}{2\sigma_i^2}\Bigr), \] and the physics-informed residual is assembled into a linear system \(H(\theta)c=r\). Instead of training all centers and widths directly, KAPI-ELM places a Gaussian-process surrogate on a low-dimensional hyperparameter vector \(\mathbf w\) governing the distribution \(p(\alpha^*,\sigma\mid \mathbf w)\), and alternates between least-squares solving for \(c\) and Bayesian optimization of \(\mathbf w\). The output-layer solve is closed form, \[ c_{\rm opt}=(H^TH+\beta I)^{-1}H^Tr, \] or \(H^\dagger r\) when \(\beta=0\) [2507.10241]. The reported performance reflects the method’s specialization to sharp gradients. For a 1D singularly perturbed convection–diffusion problem with \(\nu=10^{-4}\), XTFC uses 10,000 neurons for error \(\approx 10^{-3}\), whereas KAPI-ELM uses 1,275 neurons for error \(\approx 10^{-6}\). For a 2D Poisson problem with localized sharp source, KAPI-ELM with \(N_c=1600\), \(N_r=400\), \(\sigma_F=0.2\), and \(N_{\rm adap}=1\) achieves \(L^\infty\approx 10^{-4}\) in \(<100\) BO iterations and \(\sim 30\) s on CPU with \(\sim 700\) RBFs; the cited PINN baseline requires 50,000 training steps and \(\sim 1\) h for poorer error \(\sim 10^{-3}\). In inverse linear advection with \(5\%\) noise, the method converges in \(<20\) BO steps to \(a_{\rm est}=0.4986\) for true \(0.5\) [2507.10241]. Related kernelized physics-informed solvers broaden the role of kernels further. SK-PINN replaces automatic differentiation by smoothed-particle-hydrodynamics kernels \(W(x-x',h)\), using \[ f(x)\simeq \int f(x')\,W(x-x',h)\,dx' \] and discrete neighbor sums to approximate derivatives. In an inhomogeneous Poisson problem with \(m=40{,}401\), the reported training times are about \(97\) s for AD-PINN, \(67\) s for modified-AD PINN, and \(21\) s for SK-PINN; in a lid-driven cavity problem on an 8 GB RTX 3070 Ti, SK-PINN takes \(\sim 1.9\) h versus \(\sim 55\) h for mAD-PINN, a \(55\times\) speed-up, while the NTK spectra remain close [2411.02411]. Spline-PINN, in turn, uses Hermite spline kernels to interpolate a continuous field from a CNN-managed grid of coefficients, enabling physics-informed training without precomputed data and fast continuous inference on unseen domains [2109.07143]. ## 6. Partial physics, discrepancy control, regime switching, and interaction kernels A recurring concern in physics-informed modeling is how to prevent the data-driven component from overwhelming the physical one. Orthogonal discrepancy kernels address this by decomposing the dynamics as \[ f_{\rm total}(x)=f_{\rm phys}(x;\theta)+\delta(x), \] with \(f_{\rm phys}(x;\theta)=\Phi(x)^T\theta\) and a GP discrepancy \(\delta(x)\) constrained to the orthogonal complement of the span of \(\Phi\). Starting from a base kernel \(k(x,x')\), the orthogonal kernel is defined by subtracting the projection onto the physics subspace, \[ k_\perp(x,x')=k(x,x')-\Phi(x)^T H^{-1}\Phi(x'), \] with \(H=\int_{\mathcal X}\Phi(x)\Phi(x)^Tdx\). In the cubic-oscillator example, the orthogonal-kernel GP plus SINDy obtains NMSE \(\simeq 3.9\times 10^{-5}\) versus \(\simeq 1.0\times 10^{-3}\) for a non-orthogonal GP when the dictionary is complete, and \(\simeq 1.4\times 10^{-5}\) versus \(\simeq 8.9\times 10^{-5}\) when the \(y^3\) term is omitted. The stated interpretation is that the orthogonal GP prevents the discrepancy from “stealing” physics already explainable by \(\Phi\) [2606.21199]. A closely related structured view appears in kernel-based nonlinear system identification. There the model is \[ y_t=f_{\rm phys}(x_t;\theta)+g_K(x_t)+e_t,\qquad g_K(x)=\sum_{i=1}^T \alpha_i K(x,x_i), \] and \((\theta,\alpha)\) are fitted jointly by minimizing \[ \|Y-F(\theta)-K\alpha\|_2^2+\lambda\,\alpha^T K\alpha. \] Alternating minimization yields \[ \alpha=(K+\lambda I)^{-1}(Y-F(\theta)), \] while latent-state cases are handled by a forward UKF and a backward Unscented Rauch–Tung–Striebel smoother before the same kernel regression is applied. On a polynomial/sinusoid toy problem, the reported RMSEs are \(0.010\) for ordinary least squares, \(0.007\) for a two-step discrepancy model, and \(0.005\) for the joint PIK formulation. On the cascade-tank benchmark, simulation RMSE drops from about \(0.37\) to \(0.17\) on training data and \(0.37\) to \(0.18\) on validation, with fit increasing from \(83\%\) to \(92\%\) [2509.07634]. Physically-informed change-point kernels solve a different balance problem: the validity of the physical prior may depend on regime. These models decompose the covariance into physics-based and flexible terms and gate them with a logistic function \[ s(Z;a,x_0)=\frac{1}{1+\exp[-a(Z-x_0)]}. \] With \(K_\sigma(Z,Z')=s(Z)s(Z')\), the total covariance is \[ k_{\rm total}((Z,X),(Z',X'))=K_\sigma(Z,Z')\,k_{\rm phys}(X,X') +K_\sigma(-Z,-Z')\,k_{\rm data}(X,X'). \] In the Tamar bridge case, a lift-force polynomial kernel \(k_{\rm lift}(U,U')=\sigma_L^2(UU'+c)^2\) is gated by sigmoids in wind direction and wind speed; in the aircraft wing case, a two-mode SDOF kernel is gated by rudder angle and added to an always-on squared-exponential kernel. The wing study reports interpolation and upsampling from \(64\) Hz to \(128\) Hz, with NMSE falling from \(22.9\%\) to \(2.9\%\), greatly improved MSLL, and interpretability of the regime in which the wing oscillator engages [2506.11625]. The detector-filtering literature extends the same “kernel encodes known propagation law” idea beyond regression. In the Ising noise filter, one assigns binary spins \(s_i\in\{+1,-1\}\) to hits and minimizes \[ E(\{s_i\})=-\sum_{i