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PIKFNO: An Interpretable Neural Operator Based on Physics Informed Kernel Function
Yuan Guo, Hanshu Chen, Zhuojia Fu
TL;DR
Neural operators often lack interpretability and explicit physical constraints. PIKFNO addresses this by embedding physics-informed kernel functions into the operator architecture, achieving high accuracy, improved interpretability, and strong generalization with limited training data.
Problem
Neural operators’ limited interpretability and lack of explicit physical constraints hinder their broader scientific and engineering adoption.
Method
PIKFNO embeds learnable or analytically transformed physics-informed kernel functions into the neural operator architecture.
Results
PIKFNO achieves high predictive accuracy, improved interpretability, and superior generalization under limited training data, while PIKFNO_v2 converges fastest.
Takeaways & Limitations
PIKFNO provides a physically informed and interpretable neural-operator framework aligned with kernel expansions used in meshless methods.
Takeaways & Limitations
Meshless collocation methods face source-point placement challenges that can produce ill-conditioned coefficient matrices and compromise numerical stability.
Abstract
from arXiv · showhide
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.
1 Introduction
The introduction identifies limited interpretability and weak physical constraint integration as major obstacles for neural operators in scientific applications. It proposes PIKFNO, which uses physics-informed and learnable kernel functions to combine mathematical transparency, neural expressiveness, and operator-level physical consistency.
- Motivation: Neural operators learn mappings between infinite-dimensional function spaces, unlike traditional numerical schemes based on discretized grids or predefined basis functions.The introduction contrasts neural operators with finite element, finite difference, and meshless methods.
- Motivation: Limited interpretability obstructs broader neural-operator adoption because deep networks learn implicit representations without clear correspondence to traditional operator structures.The introduction also associates missing physical constraints with large data requirements, poor extrapolation, physical-law violations, and noise sensitivity.
- PIKFNO framework: Embedding physical priors at the operator level is intended to improve physical fidelity and generalization compared with traditional neural operator methods.The introduction distinguishes this operator-level construction from physics-informed approaches that typically impose constraints through soft loss terms.
- PIKFNO framework: PIKFNO constructs operator structures from physics-informed kernel functions that encode governing-PDE characteristics and remain consistent with kernel integral formulations.This construction explicitly incorporates physical constraints into the operator rather than relying only on implicit representations.
- PIKFNO framework: Learnable kernel parameters preserve the interpretability of classical kernel methods while providing neural networks’ expressive power.The approach is motivated by the mathematical transparency of RBF, meshless collocation, and Mercer-type kernel expansions.
2 Methodology
PIKFNO embeds physics-informed kernel functions into neural operators by linking DeepONet’s coefficient–basis structure with meshless collocation methods. It combines physical priors with data-driven learning to improve interpretability, reduce training-data requirements, and accelerate convergence.
- Operator learning: PDE solution processes are operators mapping input functions, such as coefficients, sources, or boundary conditions, to output solution functions.For time-dependent problems, the operator maps an initial state to future states.
- DeepONet limitation: DeepONet combines branch-generated coefficients with trunk-learned basis functions, but its internal representations lack explicit physical meaning and remain largely a black box.Its coefficient–basis expansion has a natural structural resemblance to kernel expansions used in meshless collocation methods.
- Physics-informed kernels: Meshless collocation methods use kernels determined by the governing PDE, whose structure directly reflects the equation’s physical information.These kernels form basis expansions analogous to operator-learning representations, including fundamental solutions such as the Laplace kernel.
- PIKFNO design: PIKFNO explicitly incorporates physics-informed kernels into neural operators, enabling kernels to be learned from data when analytical forms are difficult or unavailable.The learned kernel can be regarded as a nonsingular fundamental solution, retains implicit physical structure, and can be used independently of PIKFNO.
- Reported advantages: PIKFNO provides stronger interpretability, comparable accuracy with significantly fewer training samples, and faster convergence than traditional DeepONet.Its design integrates physical priors with data-driven learning to construct efficient, robust, and physically consistent neural operators.
3 Numerical examples and discussions
Numerical experiments on a two-dimensional Laplace equation show that PIKFNO variants achieve accurate operator prediction while exposing interpretable kernel structures. The learned PIKFNO_v1 kernel can serve as a nonsingular fundamental solution, whereas PIKFNO_v2 gains an accuracy advantage over DeepONet when training data are limited.
- Experimental setup: The experiments learn the mapping from prescribed Dirichlet boundary conditions to Laplace-equation solutions on the unit-square domain using an operator-learning framework.The dataset contains 10,000 randomly generated boundary conditions and corresponding numerical solutions, split equally into training and testing sets at 40 × 40 resolution.
- Model comparison: DeepONet, PIKFNO_v1, and PIKFNO_v2 all produce small, relatively close test errors, with PIKFNO_v1 using fewer learnable parameters and PIKFNO_v2 having the largest test error.DeepONet is most accurate because its trunk network spans a broader function space; PIKFNO_v1 learns in a radial basis function space, while PIKFNO_v2 has the most limited expressive capacity.
- Model comparison: Under a representative boundary condition, all three models closely match the reference solution with uniformly small errors, while DeepONet and PIKFNO_v1 show favorable training-loss evolution.Figure 2 compares the reference solution, predictions, absolute-error distributions, and training losses.
- Kernel validation: Using the learned PIKFNO_v1 kernel as the fundamental solution in NN-based MFS produces a numerical solution that agrees extremely well with the exact solution, with negligible errors.The verification uses the analytical solution u(x,y) = x^3 − 3xy on a disk centered at (0.5,0.5) with radius 0.5.
- Limited-data performance: With small training sets, PIKFNO_v2 achieves significantly higher prediction accuracy than DeepONet by exploiting fundamental-solution structure as prior knowledge.The comparison fixes the test set at 5000 samples and evaluates both models across different training-set sizes.
4 Conclusions
PIKFNO embeds physics-informed kernel functions into neural operator architectures, integrating physical priors with data-driven learning while aligning operator structure with meshless kernel expansions.
- 4 Conclusions: PIKFNO embeds physics-informed kernel functions directly into the neural operator architecture.This enables a principled integration of physical priors and data-driven learning.
- 4 Conclusions: PIKFNO aligns its operator structure with kernel expansions used in meshless methods, enhancing interpretability.This alignment connects the neural operator formulation with established meshless-method representations.
- 4 Conclusions: PIKFNO maintains high predictive accuracy even with limited training data.The conclusion identifies limited-data predictive performance as a notable advantage over traditional neural operators.