The universal consistency of PIKS is established for linear differential constraints, proving that for universal kernels (such as Gaussian or Mat\'ern), the estimator asymptotically learns the target while satisfying physical constraints.
Abstract
Physics-informed machine learning incorporates physical principles --often expressed via differential operators-- into data-driven models. While physics-informed neural networks (PINNs) dominate empirical applications, the complexity of neural network architectures and optimization landscapes hinders the development of a corresponding learning theory. In turn, kernel methods offer an appealing alternative with closed-form solutions and analytical tractability, yet existing guarantees primarily cover the well-specified setting where the target belongs to the native Reproducing Kernel Hilbert Space (RKHS). This imposes unrealistic regularity assumptions that physical targets often fail to satisfy. In this paper, we introduce and analyze Physics-Informed Kernel methodS (PIKS). We establish the universal consistency of PIKS for linear differential constraints, proving that for universal kernels (such as Gaussian or Mat\'ern), the estimator asymptotically learns the target while satisfying physical constraints. We further derive finite-sample bounds under suitable source conditions. Our analysis is based on extending classical operator-theoretic analysis of kernel methods to physics-informed machine learning. Numerical experiments demonstrate that PIKS can be competitive with PINNs and traditional finite element methods.
Many modern machine learning models can be understood as kernel-based function-space models, including Gaussian processes and neural tangent kernels. In scientific machine learning, differential operators are increasingly used to encode physical structure directly into such models. However, it has remained unclear under which conditions these constructions are valid machine learning models, i.e. preserve positive semi-definiteness, and whether observed instabilities arise from ill-posed modeling or numerical effects. Here, we establish a simple and sufficient condition for positive semi-definiteness: for linear differential operators of order m, the base kernel must be m-times continuously differentiable. Crucially, this guarantee holds for operators with non-constant and even discontinuous coefficients. Examples are ubiquitous in physical systems, including diffusion, material elasticity, wave propagation in inhomogeneous media, and quantum systems. We conclude that remaining instabilities are attributable to numerical issues, providing a unifying validity foundation for operator-informed kernel methods.
This work introduces a variational boosting framework in which solutions are constructed additively in function space and separates global nonlinear refinement into a sequence of well-conditioned subproblems while preserving the full variational structure of the operator.
This work proposes a new interpretable neural operator framework, termed the Physics Informed Kernel Function Neural Operator (PIKFNO), which explicitly incorporates physics informed kernel functions derived from governing equations into the neural operator architecture. Unlike traditional neural operators such as DeepONet, which rely on deep networks to implicitly learn basis functions, PIKFNO constrains the trunk network through physics informed kernel functions, thereby aligning its operator structure with the kernel expansions used in meshless collocation methods. Two construction strategies are introduced: one learns kernel functions directly from data, where the learned kernel can be regarded as a nonsingular fundamental solution, while the other builds them through transformations of analytical fundamental solutions. Numerical experiments demonstrate that PIKFNO achieves high predictive accuracy with substantially improved interpretability and superior generalization under limited training data. The proposed framework offers a new pathway for developing efficient, physically consistent, and interpretable neural operators.
The results demonstrate that PINN achieves more accurate and stable full-field vibration reconstructions than conventional PINNs, particularly under conditions involving high-frequency modes, and highlights the potential of hybrid data-physics neural frameworks as an efficient and reliable approach for solving complex PDE-governed dynamical systems.
Hailong Liu, S. Hedayatrasa, Yunpeng Zhu et al.· e-Journal of Nondestructive...· 0 citations
Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric $A(x)=\Lambda(x)U(x)$. The diagonal factor $\Lambda(x)$ controls anisotropic scaling, while the orthogonal factor $U(x)$ is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.
V. Nazarenko, T. Lidzhiev, A. Tarakanov· 0 citations
This work embeds feature interaction modules derived from factorization machines (FMs) into physics-informed neural networks (PINNs) and neural operator learning, to enhance model expressiveness for solution manifolds of parameterized partial differential equations (PDEs). Motivated by the second-order Taylor expansion of multivariate functions to characterize variable couplings, we first propose FM-PINN. It explicitly captures spatio-temporal variable interactions and improves the approximation accuracy for smooth high-order PDEs. We further group spatial coordinates, time, physical parameters, and initial and boundary conditions into independent feature sets and model their cross-group interactions. Based on this strategy, we develop FM-Operator and FM-DeepONet, which are particularly effective for nonlinear conservation laws and problems with sharp gradients or discontinuities, while offering no consistent advantage on smooth operator learning benchmarks. Numerical tests demonstrate that the proposed mechanism delivers substantial accuracy gains on challenging shock-dominated equations, indicating a promising direction for physics-consistent modeling of parameterized PDEs with strong cross-field dependencies.