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 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.
Joachim Bona-Pellissier, Giacomo Meanti, Matteo Santacesaria et al.· arXiv.org· 1 citation
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.
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.
Modern engineering simulations frequently operate in regimes where
explicit
governing equations are incomplete or experiments are prohibitively expensive. To remain accurate and trustworthy under such constraints, we propose the
Physics‐Informed Neural Network Architecture (PINNA)
, a deep‐learning surrogate that fuses data with domain knowledge
inside
the network. Concretely, a fully connected
encoder
embeds raw inputs (e.g., material descriptors, loads, or operating conditions) into a latent space, while an
intermediate‐physics head
is
explicitly supervised
to predict expert‐selected quantities with clear physical meaning, such as strain‐energy densities or chemically relevant indicators. These intermediate predictions are
residual‐concatenated
with the original inputs and passed to a
task decoder
, enabling the network to learn and correct mismatches between approximate physics and observed responses. This architecture achieves (i) higher
accuracy
and sample efficiency than purely data‐driven baselines, (ii) negligible
computational overhead
at inference time, and (iii) intrinsic
interpretability
, as intermediate predictions expose physically meaningful internal representations. We further introduce a scalable extension, Generalized PINNA (
G‐PINNA
), which stacks multiple physics heads to accommodate multiscale and multi‐physics constraints within the same residual‐concatenation framework. PINNA is validated across three fundamentally different benchmarks: two composite‐material problems involving nonlinear stress‐strain behavior and multistage failure, and a large‐scale 1‐D laminar combustion problem governed by stiff chemical kinetics, thermal transport, and reduced fluid mechanics. In the composite benchmarks, PINNA reduces test errors by up to an order of magnitude relative to Fourier Neural Operators and DeepONets while using fewer parameters. In the combustion benchmark, PINNA accurately predicts both interpretable intermediate chemical indicators and high‐dimensional flame quantities, including scalar metrics and full spatial profiles, demonstrating that intermediate supervision enables robust generalization beyond solid mechanics. These results confirm that embedding expert knowledge directly into neural architectures provides a practical, interpretable, and general framework for learning implicit physics across diverse engineering domains.
Zheng-Tao Yao, Philippe Hawi, V. Aitharaju et al.· International Journal for Nu...· 0 citations
It is concluded that remaining instabilities are attributable to numerical issues, providing a unifying validity foundation for operator-informed kernel methods.
This work proposes a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs that achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.
Pinki Khatun, M. Sajid, Abhinav Jha et al.· arXiv.org· 0 citations
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