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Evaluation of Sampling Strategies and Physics-Informed Kolmogorov--Arnold Networks in Unbounded Domains

This paper evaluates physics-informed learning strategies for inverse PDE problems in unbounded domains, demonstrating that Gaussian and exponential sampling distributions enhance far-field reconstruction accuracy while Kolmogorov–Arnold Networks (PIKANS) achieve comparable or superior accuracy with fewer neurons than traditional MLPs, albeit at a higher computational cost.

Original authors: Gregorio Pérez-Bernal, Oscar Rincón-Cardeño, Silvana Montoya-Noguera, Nicolás Guarín-Zapata

Published 2026-10-05

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CC BY 4.0

Original authors: Gregorio Pérez-Bernal, Oscar Rincón-Cardeño, Silvana Montoya-Noguera, Nicolás Guarín-Zapata

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to understand a vast, invisible landscape by taking measurements at a few scattered points. This is the daily challenge for scientists who study phenomena that stretch endlessly, such as the way heat dissipates into the open air or how seismic waves travel through the Earth's crust. In these "unbounded" worlds, there are no walls to stop the action, and the effects of a disturbance can theoretically ripple out forever. To study such things on a computer, researchers must build digital models that mimic these infinite spaces. However, a computer cannot hold an infinite amount of information. It must choose a finite number of spots to sample, and if it chooses those spots poorly—perhaps by spreading them out evenly across a vast, empty area—it might miss the most important details where the action is actually happening.

For decades, scientists have used a powerful tool called a physics-informed neural network to solve these puzzles. Think of this tool as a smart student that learns the rules of physics, like the laws governing heat or fluid flow, and then tries to guess the solution to a specific problem. The student is given a few real-world measurements and a set of physical laws, and it must figure out the rest of the picture. But when the problem involves an infinite space, this student often struggles. It tends to focus too much on the center of the map and forgets how the solution behaves far away, or it gets confused by the sheer size of the empty space it is trying to fill.

A team of researchers at Universidad EAFIT in Colombia recently set out to fix this specific problem. They wanted to know if they could teach these digital students to pay attention to the right places and if a newer, more flexible type of computer brain could help them do it better. They tested two main ideas: first, changing how the computer picks its sample points, and second, swapping the standard computer brain for a newer design that learns differently.

The researchers began by looking at how the computer chooses where to look. In a standard approach, the computer picks points evenly, like a grid of dots spread across a field. The team realized that for problems where the solution fades away as you move further out, this even spread is wasteful. Instead, they tried teaching the computer to pick more points where the action is happening and fewer points where things are quiet. They used two specific patterns for this: one that clusters points around the center like a bell curve, and another that places many points near a boundary and then lets them thin out rapidly as you move away. They found that this simple change made a huge difference. By concentrating their attention where the solution changed the most, the computer models became much better at guessing what was happening in the far reaches of the domain, even in areas where they had no direct data. It was as if the student learned to focus their study time on the most difficult chapters of a book rather than skimming the whole thing equally.

Next, the team compared the standard computer brain, known as a multilayer perceptron, with a newer architecture called a Kolmogorov–Arnold network. The standard brain works by passing information through a series of fixed filters that cannot change their shape. The newer brain, however, is built differently. Instead of fixed filters, it uses flexible, learnable curves on the connections between its parts. This allows it to adapt its shape to fit the specific problem it is solving. The researchers found that this newer brain was incredibly efficient. It could achieve the same level of accuracy as the standard brain but with far fewer internal components. It was like finding a way to solve a complex puzzle with fewer pieces.

However, there was a catch. While the newer brain was more efficient in terms of the number of parts it used, it was slower to work. Because its flexible curves had to be calculated and adjusted constantly, it took more time to train and more time to make predictions. The standard brain was faster, even if it needed more parts to get the job done. The researchers concluded that the choice between the two depends on what is more important for the specific task: if speed is the priority, the standard brain is better; if accuracy with a smaller model is the goal, the newer brain is the superior choice.

The study also highlighted that these improvements were most noticeable when the computer was asked to predict what happens far away from the known data. In the unbounded domains they tested, the ability to correctly guess the behavior of the solution in the distance was the true test of success. By combining the smarter way of picking sample points with the more flexible brain architecture, the researchers showed that it is possible to build models that understand the infinite without needing to artificially cut off the world. Their work suggests that for scientists trying to model the endless reaches of the physical world, the key lies not just in having more data, but in teaching the computer how to look at the data it has.

Technical Summary: Evaluation of Sampling Strategies and Physics-Informed Kolmogorov–Arnold Networks in Unbounded Domains

Problem Statement
Physics-Informed Neural Networks (PINNs) have become a standard framework for solving partial differential equations (PDEs) and inverse problems by embedding physical laws into the learning process. However, their application to unbounded domains (infinite and semi-infinite regions) remains challenging. The primary difficulty lies in representing an unbounded region with a finite set of training points while preserving the solution's asymptotic behavior. Traditional numerical methods often address this by truncating the domain and imposing artificial boundary conditions, which can introduce errors dependent on the truncation location. Furthermore, inverse problems in these domains require the simultaneous approximation of the solution field and the identification of unknown spatially varying parameters, creating a complex optimization landscape.

Methodology
The authors propose a systematic evaluation of complementary strategies to improve PINN performance in unbounded inverse problems. The framework integrates three key components:

Dual-Network Architecture: Instead of a single multi-output network, the framework employs two independent neural networks: one ( $\mathcal{N}_u$ ) to approximate the solution field $u(\mathbf{x})$ and another ( $\mathcal{N}_k$ ) to infer the unknown coefficient field $k(\mathbf{x})$ . These networks are coupled through a physics-informed loss function derived from the governing Poisson equation ( $-\nabla \cdot (k \nabla u) = f$ ).
Physics-Informed Sampling Strategies: To address the unbounded nature of the domain without arbitrary truncation, the study evaluates non-uniform sampling distributions that incorporate prior knowledge of the solution's asymptotic decay.
- Uniform Sampling: Standard sampling over a finite box.
- Gaussian Sampling: Used for infinite domains, concentrating points around the region of interest while maintaining a non-zero probability density over the unbounded domain.
- Exponential Sampling: Used for semi-infinite domains (combined with Gaussian for the other dimension), placing more points near the boundary and progressively fewer points toward the asymptotic region.
Network Architectures (MLP vs. KAN): The study compares conventional Multilayer Perceptrons (MLPs), which use fixed activation functions at nodes, against Kolmogorov–Arnold Networks (KANs). KANs are grounded in the Kolmogorov–Arnold representation theorem, replacing fixed node activations with learnable spline-based functions on the network edges. This allows the network to adapt its functional representation during training.

The evaluation utilizes manufactured solutions for two benchmark inverse problems: one on an infinite domain and one on a semi-infinite domain. Both involve recovering a spatially varying coefficient $k$ and the solution $u$ from sparse observational data and the governing PDE.

Key Contributions

Systematic Benchmarking: The paper provides a quantitative assessment of how sampling distributions (Uniform, Gaussian, Exponential) affect reconstruction accuracy and generalization in unbounded inverse problems.
PIKAN Evaluation: It introduces and evaluates Physics-Informed KANs (PIKANs) for inverse problems in unbounded domains, a combination not previously evaluated in this context.
Trade-off Analysis: The study characterizes the trade-offs between predictive accuracy, model size (number of neurons vs. parameters), and computational cost (training time, inference, and derivative evaluation) between MLPs and PIKANs.
Adaptive Weighting: The framework employs adaptive loss weighting to balance the competing objectives of reconstructing the solution field and inferring the coefficient field.

Results

Sampling Strategies: Gaussian and exponential sampling strategies significantly outperform uniform sampling, particularly in the far field. By concentrating collocation points where the solution exhibits significant variation and decaying toward the asymptotic region, these strategies improve reconstruction accuracy without requiring explicit domain truncation. The distribution parameters ( $\sigma$ for Gaussian, $\lambda$ for exponential) allow for direct embedding of problem-specific prior knowledge.
Accuracy vs. Efficiency:
- Accuracy: PIKANs match or exceed the accuracy of MLP-based PINNs. They achieve comparable or lower global errors while requiring fewer hidden neurons.
- Parameters & Cost: Despite using fewer neurons, PIKANs do not necessarily reduce the total number of trainable parameters or computational cost. The learnable spline functions on the edges introduce additional parameters (spline coefficients, scaling factors) and increase the computational overhead for forward passes and, notably, for derivative evaluation (required for PDE residuals).
- Training Time: PIKANs require more time to train and perform inference compared to MLPs under identical conditions. The cost is most pronounced for second-order derivatives.

Significance and Claims
The paper claims that the proposed methodology offers a direct mechanism for embedding prior knowledge about the spatial behavior of solutions into the training process, bypassing the need for artificial domain truncation. The findings suggest that while PIKANs offer superior approximation flexibility (achieving high accuracy with fewer neurons), this comes at a higher computational cost due to the complexity of spline evaluation and differentiation.

The authors position this work as a controlled investigation into the effects of sampling and architecture on unbounded inverse problems. They modestly note that while the benchmarks are simplified, they isolate specific methodological choices to guide model design. The paper concludes that future work should focus on reducing the computational overhead of spline evaluation to make PIKANs viable for large-scale applications and exploring hybrid formulations (e.g., Boundary Element Methods combined with PINNs) to handle radiation conditions more rigorously. The study does not claim universal superiority for KANs but highlights their potential when the flexibility of learnable edge functions outweighs the computational penalty.

Technical Summary: Evaluation of Sampling Strategies and Physics-Informed Kolmogorov–Arnold Networks in Unbounded Domains

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