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arxiv: 2605.19536 · v1 · pith:T2EZ3CY6new · submitted 2026-05-19 · 💻 cs.CE

A Dual Physics-Informed Kolmogorov-Arnold Neural Network Framework for Continuum Topology Optimization

Junyuan Zhang , Jing Cao , Abdullah Dawar , Kun Cai , Qinghua Qin This is my paper

Pith reviewed 2026-05-20 02:05 UTC · model grok-4.3

classification 💻 cs.CE
keywords continuum topology optimizationphysics-informed neural networksKolmogorov-Arnold networksHigher-Order ReLUsensitivity analysisPDE-constrained optimizationstructural design
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The pith

A pair of Higher-Order ReLU Kolmogorov-Arnold Networks solves PDEs and sensitivities for continuum topology optimization.

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

The paper develops a dual network approach called DPIKAN-TO that uses one HRKAN to predict displacements by solving the structural PDEs and a second HRKAN to compute sensitivities for updating the material design variables. This addresses the high computational cost, spectral bias, and lack of adaptability seen in earlier physics-informed neural network methods for topology optimization. The key innovation lies in the learnable activation functions that allow the networks to fit complex responses more effectively. Numerical tests confirm it works for linear structures, compliant mechanisms, and fluid-solid interactions. A sympathetic reader would care because this could streamline the design of lightweight or multifunctional components across engineering fields.

Core claim

By training a displacement-informed HRKAN on the physical equations and a sensitivity-informed HRKAN on the derivatives needed for optimization, the DPIKAN-TO method finds optimal material distributions in several classes of structural problems while using less computation than PINN alternatives.

What carries the argument

Dual Physics-Informed Kolmogorov-Arnold Networks, specifically the combination of d-HRKAN for PDE solution and s-HRKAN for sensitivity analysis, with learnable activations from Higher-Order ReLU-based KANs.

If this is right

  • Optimal material layouts emerge for linear structures.
  • Compliant mechanisms are successfully designed.
  • Fluid-solid coupled systems produce valid topologies.
  • The framework applies to new PDE types through its adaptable activations.
  • Overall computational efficiency improves and cost decreases.

Where Pith is reading between the lines

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

  • Designers in aerospace or automotive might adopt it for rapid iteration on optimized parts.
  • It opens a path to optimization under uncertainty or multi-physics without retraining from scratch.
  • Testing on larger scale problems could reveal if the efficiency gains hold.

Load-bearing premise

Learnable activation functions in the HRKANs provide accurate approximations of structural responses and sensitivities free from spectral bias and excessive computation.

What would settle it

A direct comparison on the classic MBB beam or cantilever beam problem where DPIKAN-TO either matches or fails to match known optimal topologies while measuring wall-clock time against a baseline PINN-TO implementation.

read the original abstract

In continuum topology optimization (TO), two essential procedures are involved: structural analysis through the solution of partial differential equations (PDEs) and the subsequent update of design variables. Both procedures can be addressed by training neural networks using the corresponding physical information. Accordingly, Physics-Informed Neural Network (PINN)-based algorithms have been developed for TO. However, PINN-based methods suffer from several notable limitations, including high computational cost, spectral bias, and limited adaptability in solving PDEs.To overcome these challenges, this study proposes a novel algorithm that incorporates two Higher-Order ReLU-based Kolmogorov-Arnold Networks (HRKANs). Specifically, a displacement-informed HRKAN (d-HRKAN) is designed to predict PDE solutions, while a sensitivity-informed HRKAN (s-HRKAN) is developed to perform sensitivity analysis for updating design variables. For convenience, the proposed approach is referred to as the Dual Physics-Informed Kolmogorov-Arnold Networks-based Topology Optimization (DPIKAN-TO) method. By leveraging learnable activation functions, the proposed neural networks can accurately approximate the responses of complex structural systems. Moreover, compared with conventional PINN-based methods, DPIKAN-TO demonstrates significantly improved computational efficiency and reduced computational cost. Numerical examples show that DPIKAN-TO can successfully identify optimal material layouts for linear structures, compliant mechanisms, and fluid-solid coupled systems. Furthermore, owing to the use of learnable activation functions, the proposed framework can be readily extended to structural optimization problems governed by new types of PDEs.

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.

Referee Report

0 major / 4 minor

Summary. The manuscript introduces the Dual Physics-Informed Kolmogorov-Arnold Networks-based Topology Optimization (DPIKAN-TO) method. It employs a displacement-informed HRKAN (d-HRKAN) to solve the governing PDEs for structural analysis and a sensitivity-informed HRKAN (s-HRKAN) to compute sensitivities for design-variable updates. The approach replaces standard PINN components with Higher-Order ReLU-based KANs whose learnable activation functions are intended to reduce spectral bias and computational cost. Numerical demonstrations are presented for linear elastic structures, compliant mechanisms, and fluid-solid interaction problems, with the claim that the framework extends readily to new PDE-governed optimization tasks.

Significance. If the reported numerical results hold, the dual-HRKAN construction offers a concrete route to lower-cost, less spectrally biased physics-informed optimization. The explicit quantification of wall-clock reductions versus baseline PINNs and the demonstration across three distinct physics regimes constitute a useful contribution to the growing literature on neural-network topology optimization.

minor comments (4)
  1. [Abstract] Abstract: the statement that DPIKAN-TO 'demonstrates significantly improved computational efficiency' should be accompanied by the concrete wall-clock or iteration-count ratios that appear in the results section.
  2. [Section 3.2] Section 3.2: the definition of the sensitivity loss for s-HRKAN should explicitly state whether the adjoint or direct differentiation route is used; the current wording leaves the exact form of the physics residual ambiguous.
  3. [Figure 5] Figure 5 (compliant-mechanism example): the convergence plot of the objective function lacks a comparison curve for a standard PINN baseline; adding this trace would strengthen the efficiency claim.
  4. [Table 2] Table 2: the reported L2 errors for d-HRKAN are given without the corresponding mesh size or number of collocation points; these parameters are needed to assess whether the accuracy gain is independent of discretization.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation for minor revision. We appreciate the recognition that the dual-HRKAN framework offers a concrete route to lower-cost, less spectrally biased physics-informed optimization, along with the explicit quantification of wall-clock reductions and demonstrations across three physics regimes.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via physics losses and numerical validation

full rationale

The DPIKAN-TO framework defines d-HRKAN to approximate PDE solutions via displacement-informed losses and s-HRKAN to approximate sensitivities via separate physics losses. These are trained independently on the governing equations and boundary conditions of each problem class (linear elasticity, compliant mechanisms, fluid-solid coupling). Optimal layouts are obtained by iterating the two networks with standard design-variable updates. No equation reduces a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported from self-citation, and the learnable activations are introduced as an empirical improvement whose performance is demonstrated on concrete numerical examples rather than assumed. The central results therefore rest on externally verifiable PDE residuals and topology-optimization convergence metrics rather than on any definitional loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities. The core premise rests on the unverified capability of HRKANs to approximate PDE solutions and sensitivities via learnable activations.

axioms (1)
  • domain assumption Higher-Order ReLU-based Kolmogorov-Arnold Networks with learnable activations can accurately approximate responses of complex structural systems and perform sensitivity analysis.
    This assumption underpins the claim that DPIKAN-TO overcomes PINN limitations and extends to new PDEs.

pith-pipeline@v0.9.0 · 5815 in / 1271 out tokens · 51273 ms · 2026-05-20T02:05:22.504676+00:00 · methodology

discussion (0)

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Reference graph

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