Solving Nonlinear PDEs with Sparse Radial Basis Function Networks

Konstantin Pieper; Xiaochuan Tian; Zihan Shao

arxiv: 2505.07765 · v3 · submitted 2025-05-12 · 🧮 math.NA · cs.LG· cs.NA

Solving Nonlinear PDEs with Sparse Radial Basis Function Networks

Zihan Shao , Konstantin Pieper , Xiaochuan Tian This is my paper

Pith reviewed 2026-05-22 15:56 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NA

keywords sparse radial basis function networksnonlinear PDEsreproducing kernel Banach spacesrepresenter theoremcollocation methodsphysics-informed neural networks

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The pith

A representer theorem proves that sparse radial basis function optimization in reproducing kernel Banach spaces yields finite solutions for nonlinear PDEs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a sparse RBF network approach for nonlinear PDEs that uses regularization to enforce sparsity and limit redundant neurons. It places the method inside reproducing kernel Banach spaces generated by one-hidden-layer networks that may have infinite width. A representer theorem then shows the sparse optimization problem always admits a finite solution and supplies error bounds. This construction is meant to combine the reliability of classical collocation with the flexibility of neural approximations while avoiding over-parameterization. A three-phase algorithm implements the theory through adaptive feature selection, second-order optimization, and neuron pruning.

Core claim

The central discovery is a representer theorem in the RKBS induced by one-hidden-layer neural networks of possibly infinite width: the sparse optimization problem admits a finite solution, and the associated error bounds supply a foundation for generalizing classical numerical analysis to this sparse RBF collocation setting for nonlinear PDEs.

What carries the argument

Reproducing kernel Banach spaces induced by one-hidden-layer neural networks of possibly infinite width, which serve as the function space in which the sparse RBF collocation problem is well-posed and the representer theorem holds.

If this is right

The three-phase algorithm maintains computational efficiency through adaptive feature selection, second-order optimization, and pruning of inactive neurons.
Numerical experiments show the method is effective and offers advantages over Gaussian process approaches in selected cases.
Error bounds derived from the representer theorem provide a basis for generalizing classical numerical analysis techniques to the sparse RBF setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same RKBS representer argument may extend to other kernel-based collocation schemes beyond radial basis functions.
Automatic adaptation of network width during solution could reduce the need for manual tuning of basis functions in high-dimensional PDEs.
The framework might transfer error-analysis tools from finite-width to infinite-width network models in related approximation problems.

Load-bearing premise

The reproducing kernel Banach space induced by one-hidden-layer neural networks of possibly infinite width supplies the correct setting in which the sparse RBF collocation problem is well-posed and admits a finite representer.

What would settle it

A concrete counter-example in which the sparse optimization problem formulated in this RKBS has no finite solution for a standard nonlinear PDE such as Burgers' equation, or in which the stated error bounds are violated by direct numerical computation, would falsify the claim.

Figures

Figures reproduced from arXiv: 2505.07765 by Konstantin Pieper, Xiaochuan Tian, Zihan Shao.

**Figure 2.** Figure 2: Numerical results using Sparse RBFNet (our) and Gaussian Process (GP) on [PITH_FULL_IMAGE:figures/full_fig_p035_2.png] view at source ↗

**Figure 3.** Figure 3: Numerical results of different treatments of boundary conditions. (a) Exact [PITH_FULL_IMAGE:figures/full_fig_p039_3.png] view at source ↗

**Figure 4.** Figure 4: Numerical results of regularized Eikonal equation ( [PITH_FULL_IMAGE:figures/full_fig_p040_4.png] view at source ↗

**Figure 5.** Figure 5: Convergence of numerical solution to viscosity solution [PITH_FULL_IMAGE:figures/full_fig_p041_5.png] view at source ↗

**Figure 6.** Figure 6: Viscous Burgers equation solved by Sparse RBFNet coupled with fully implicit [PITH_FULL_IMAGE:figures/full_fig_p042_6.png] view at source ↗

**Figure 7.** Figure 7: Convergence of residual loss for α = 10−3 with/without pretraining. and ∇2 ˆℓ ≈ JRW JR + Cor A.2.2 Line search As is customary in Gauss-Newton algorithm, we do a line search of optimal step size. We shrink step size θ, by 2/3 each time, starting from θ = 1, until the actual descent is within trust-region, which is set to be Actual Descent ≤ h × Estimated Descent = hθGT (DPz). where h ∈ [1/5, 1/3]. In all n… view at source ↗

**Figure 8.** Figure 8: Small-viscosity Burgers equation (ν = 0.002): comparison of solution slices with different numbers of collocation points (K). All results are obtained using time step ∆t = 0.01 and regularization parameter α = 10−4 . (a) K = 40; (b) K = 80; (c) K = 200; (d) K = 400. The size of each ⋆ indicates the bandwidth of the corresponding kernel. Appendix D. Burgers equation with small viscosity We briefly discuss t… view at source ↗

read the original abstract

We propose a novel framework for solving nonlinear PDEs using sparse radial basis function (RBF) networks. Sparsity-promoting regularization is employed to prevent over-parameterization and reduce redundant features. This work is motivated by longstanding challenges in traditional RBF collocation methods, along with the limitations of physics-informed neural networks (PINNs) and Gaussian process (GP) approaches, aiming to blend their respective strengths in a unified framework. The theoretical foundation of our approach lies in the function space of Reproducing Kernel Banach Spaces (RKBS) induced by one-hidden-layer neural networks of possibly infinite width. We prove a representer theorem showing that the sparse optimization problem in the RKBS admits a finite solution and establishes error bounds that offer a foundation for generalizing classical numerical analysis. The algorithmic framework is based on a three-phase algorithm to maintain computational efficiency through adaptive feature selection, second-order optimization, and pruning of inactive neurons. Numerical experiments demonstrate the effectiveness of our method and highlight cases where it offers notable advantages over GP approaches. This work opens new directions for adaptive PDE solvers grounded in rigorous analysis with efficient, learning-inspired implementation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Sparse RBFs in RKBS get a representer theorem and three-phase algorithm for nonlinear PDEs, but the nonlinear residual may need extra justification to fit the finite-solution claim.

read the letter

The main thing here is a sparse RBF network for nonlinear PDEs that uses RKBS from one-hidden-layer networks to prove a representer theorem giving finite solutions plus error bounds, paired with a three-phase adaptive algorithm for feature selection, second-order steps, and pruning. It tries to fix over-parameterization in classical RBF collocation while adding some learning-style efficiency and showing better results than GPs in the reported tests. The synthesis of sparsity regularization with the RKBS setting and the practical algorithm is the clearest new piece; it gives a concrete way to keep the basis small without losing the collocation structure. The numerics apparently demonstrate advantages in accuracy or cost for the cases they ran, which is worth noting even if the details are limited in the abstract. The representer result and bounds are presented as independent of the experiments, which is a plus for avoiding circularity. That said, the stress-test concern lands: the PDE residual for a nonlinear operator is not obviously convex or pointwise in the way many RKBS representer theorems require, so the finite-support conclusion does not transfer automatically. If the proof section does not add a specific argument for the nonlinear case, the central theoretical claim has a gap. The error bounds are asserted but their dependence on the nonlinearity or the sparsity level is not visible here, so they could be loose in practice. This is aimed at people building adaptive meshless solvers who already know RBF and PINN literature and want a version with explicit sparsity control and some analysis. A reader looking for hybrid numerical-ML methods with theory would find usable ideas. It is solid enough on the algorithmic side and specific enough on the claims to go to peer review rather than desk reject; the proofs and experiments need checking but the direction is worth referee time.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a framework for solving nonlinear PDEs via sparse radial basis function networks, set in Reproducing Kernel Banach Spaces induced by one-hidden-layer neural networks of possibly infinite width. It proves a representer theorem guaranteeing that the associated sparse optimization problem admits a finite solution, derives error bounds, and introduces a three-phase algorithm (adaptive feature selection, second-order optimization, pruning) whose effectiveness is illustrated on numerical examples that are claimed to outperform Gaussian-process baselines in selected cases.

Significance. If the representer theorem and error bounds are shown to hold for the nonlinear residual loss, the work would supply a rigorous, sparsity-promoting foundation that generalizes classical RBF collocation analysis while retaining computational advantages over dense PINN or GP formulations.

major comments (1)

[§3] §3 (Representer Theorem and RKBS setting): The proof establishes finite support for the sparse optimization problem under the assumption that the loss depends on the function through a convex combination of point evaluations or satisfies a subdifferential condition. For the actual PDE objective, which includes the residual of a nonlinear differential operator evaluated at collocation points, no auxiliary argument is supplied showing that the nonlinearity preserves the required structure. Without this verification the finite-representer conclusion does not automatically transfer to the nonlinear PDE problem.

minor comments (2)

[Algorithm section] The statement of the three-phase algorithm would benefit from an explicit description of the pruning threshold and the precise form of the second-order optimizer used in phase two.
[Notation] Notation for the RKBS norm and the sparse regularizer should be unified between the theoretical statements and the algorithmic pseudocode.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address the major comment below and outline the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [§3] §3 (Representer Theorem and RKBS setting): The proof establishes finite support for the sparse optimization problem under the assumption that the loss depends on the function through a convex combination of point evaluations or satisfies a subdifferential condition. For the actual PDE objective, which includes the residual of a nonlinear differential operator evaluated at collocation points, no auxiliary argument is supplied showing that the nonlinearity preserves the required structure. Without this verification the finite-representer conclusion does not automatically transfer to the nonlinear PDE problem.

Authors: We agree that the representer theorem as stated in §3 relies on the loss satisfying either a convex-combination-of-point-evaluations structure or a subdifferential condition, and that the manuscript does not supply an explicit auxiliary argument confirming that the nonlinear residual loss (e.g., squared residual of a nonlinear differential operator at collocation points) inherits this structure. In the revised version we will add a short lemma (or remark) after the statement of the theorem that verifies the required condition for the specific loss used in the PDE examples. Under the standing assumption that the nonlinearity is locally Lipschitz, the residual evaluated at a finite set of collocation points remains a continuous function of finitely many pointwise values of the network and its derivatives; because the network itself is a finite linear combination of kernel sections, the composite loss still satisfies the subdifferential condition employed in the proof. We will also note the precise Lipschitz or smoothness hypotheses needed on the nonlinearity so that the finite-representer conclusion carries over directly to the nonlinear setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theoretical claims are independent of numerical fits

full rationale

The paper's central derivation consists of a representer theorem and error bounds proved in the RKBS setting for the sparse optimization problem. No quoted step reduces the finite-support conclusion or the error estimates to a fitted parameter, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The abstract and description present the RKBS representer result as a self-contained mathematical statement whose validity does not presuppose the target PDE residual or the numerical experiments. The three-phase algorithm and numerical tests are presented separately as implementation details, not as inputs that define the theorem. This satisfies the default expectation that a theoretical paper with an explicit proof is non-circular unless a specific equation or citation is shown to collapse the claim by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the RKBS setting for one-hidden-layer networks and the validity of the sparse optimization problem within that space; no explicit free parameters or invented entities are stated in the abstract.

axioms (1)

domain assumption The function space of Reproducing Kernel Banach Spaces induced by one-hidden-layer neural networks of possibly infinite width is the appropriate setting for the sparse RBF collocation problem.
Stated as the theoretical foundation of the approach.

pith-pipeline@v0.9.0 · 5728 in / 1158 out tokens · 26816 ms · 2026-05-22T15:56:48.602679+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove a representer theorem showing that the sparse optimization problem in the RKBS admits a finite solution
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the function space of Reproducing Kernel Banach Spaces induced by one-hidden-layer neural networks

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

An adaptive radial basis function approach for efficiently solving multidimensional spatiotemporal integrodifferential equations
math.NA 2026-04 unverdicted novelty 5.0

An adaptive RBF approach is introduced that adjusts scales and centers to efficiently solve anisotropic spatiotemporal integrodifferential equations while remaining mesh-free.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · cited by 1 Pith paper

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work page doi:10.1137/24m1658279

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Jinshuai Bai, Gui-Rong Liu, Ashish Gupta, Laith Alzubaidi, Xi-Qiao Feng, and YuanTong Gu

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Claire Boyer, Antonin Chambolle, Yohann De Castro, Vincent Duval, Fr´ ed´ eric De Gournay, and Pierre Weiss

URLhttps://arxiv.org/ abs/2406.09217. Claire Boyer, Antonin Chambolle, Yohann De Castro, Vincent Duval, Fr´ ed´ eric De Gournay, and Pierre Weiss. On representer theorems and convex regularization.SIAM Journal on Optimization, 29(2):1260–1281,

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work page 2025

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Gregory E Fasshauer. Solving partial differential equations by collocation with radial basis functions. InProceedings of Chamonix, volume 1997, pages 1–8. Citeseer,

work page 1997

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Carsten Franke and Robert Schaback

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work page doi:10.1007/s10107-020-01530-0

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Konstantin Pieper, Zezhong Zhang, and Guannan Zhang

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work page doi:10.1051/cocv/

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Zihan Shao, Konstantin Pieper, and Xiaochuan Tian. Sparse RBF networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithms. arXiv preprint arXiv:2601.17562,

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doi: https:// doi.org/10.1016/j.cma.2021.114333

ISSN 0045-7825. doi: https:// doi.org/10.1016/j.cma.2021.114333. URLhttps://www.sciencedirect.com/science/ article/pii/S0045782521006186. Michael E Taylor.Partial differential equations III, volume

work page doi:10.1016/j.cma.2021.114333 2021

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doi: 10.1007/3-7643-7582-5

ISBN 978-3-7643-7581-2. doi: 10.1007/3-7643-7582-5. Michael Ulbrich.Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. MOS-SIAM Series on Optimization. SIAM,

work page doi:10.1007/3-7643-7582-5

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Gerd Wachsmuth and Daniel Walter

ISBN 9781611970685. Gerd Wachsmuth and Daniel Walter. No-gap second-order conditions for minimization problems in spaces of measures.arXiv preprint arXiv:2403.12001,

work page arXiv

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A mini-batch method for solving nonlinear PDEs with Gaussian processes.arXiv preprint arXiv:2306.00307,

Xianjin Yang and Houman Owhadi. A mini-batch method for solving nonlinear PDEs with Gaussian processes.arXiv preprint arXiv:2306.00307,

work page arXiv

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URLhttps://doi.org/10.1007/s10915-024-02463-y

doi: 10.1007/s10915-024-02463-y. URLhttps://doi.org/10.1007/s10915-024-02463-y. 57

work page doi:10.1007/s10915-024-02463-y