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Mesh-Free Numerical Approximation of the Biharmonic Equation via Optimized Kolmogorov--Arnold Neural Networks

B. Veena S. N. Rao

arXiv:2609.39469v1math.NAcs.NA

TL;DR

Fourth-order plate problems pose numerical challenges from derivative coupling, clamped boundary conditions, and optimization difficulties in physics-informed losses. The paper develops a mesh-free KAN-PINN with smooth activations, a direct residual formulation, and hybrid optimization. On a manufactured unit-square benchmark, automated exploration reduces relative L2 error by more than 150× to 1.593 × 10^-5.

  • Problem

    Fourth-order plate problems require careful treatment of high-order derivative coupling and multiple boundary conditions, while PINN loss optimization can be difficult for higher-order operators.

    Source: §1 · p. 2

  • Method

    The framework uses edge-based KAN functions, tanh activations, a direct normalized residual, and a hybrid AdamW-to-L-BFGS schedule for the clamped biharmonic problem.

    Source: §3.3 · p. 9, §6 · p. 15

  • Results

    An automated search reduced relative L2 error from 0.24% to 0.00159% (1.593 × 10^-5), an overall reduction exceeding 150×.

    Source: §5.4 · p. 14

  • Takeaways & Limitations

    On the manufactured benchmark, the compact 7,801-parameter KAN-PINN attained relative L2 error 1.593 × 10^-5 without explicit mesh generation or auxiliary variable transformations.

    Source: §6 · p. 15

    Classical C4(Ω) regularity on the square requires suitable forcing and boundary-data regularity and compatibility, with corner singularities requiring care.

    Source: §2.3 · p. 6

Abstract

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A mesh-free numerical framework based on \textit{Kolmogorov--Arnold Physics-Informed Neural Networks} (KAN-PINNs) is developed for the approximation of fourth-order elliptic boundary value problems, with specific application to the biharmonic equation governing thin plate deflection. Unlike conventional Multi-Layer Perceptrons relying on fixed nodal activations, learnable univariate functions parameterized via radial basis functions are deployed on network edges, while high-order differentiability is preserved through hyperbolic tangent activation mappings. The severe numerical stiffness inherent to fourth-order differential operators and fully clamped boundary conditions is addressed through a direct normalized residual formulation coupled with a hybrid, multi-stage AdamW-to-L-BFGS optimization pipeline. An automated \textit{24/7 hill-climbing search protocol} is implemented to systematically calibrate boundary penalty weights and optimization schedules. When evaluated on a smooth manufactured benchmark on the unit square, an error reduction factor exceeding $150\times$ is attained over successive iterations, culminating in a final relative $L_2$ error of $1.593 \times 10^{-5}$ ($0.00159\%$) and a training loss of $3.065 \times 10^{-6}$ utilizing only $7,801$ trainable parameters. These results demonstrate that high-order PDE problems can be accurately resolved using compact, mesh-free KAN-PINNs architecture without requiring auxiliary variable transformations or discrete mesh generation.

1 Introduction

Classical discretizations for fourth-order plate problems require careful treatment of derivatives and boundary conditions, while mesh generation and large systems can add overhead. The paper proposes a mesh-free KAN-PINN framework with a direct residual formulation and automated multi-stage optimization.

1 Introduction

  • Fourth-order plate problems require careful treatment of derivative coupling and multiple boundary conditions, while complex domains or higher dimensions can bring mesh-generation overhead and large sparse systems.Source: §1 · p. 2
  • PINNs use a neural network as a global solution approximation and automatic differentiation to evaluate PDE derivatives, enforcing equations and boundary conditions through training.This avoids constructing a conventional mesh and explicitly assembling a discrete PDE operator.Source: §1 · p. 2
  • PINN loss components can have unbalanced optimization dynamics, and challenging PDEs can expose failures arising from objective structure and optimization rather than insufficient network expressivity.These hurdles are amplified for the biharmonic equation by repeated differentiation of network parameters.Source: §1 · pp. 2–3
  • KANs place learnable univariate functions on network edges, and KINNs combine KAN architectures with physics-informed strong-form, energy-form, and inverse-form PDE formulations.The introduction situates these architectures among classical discretization, PINNs, and variational neural approaches.Source: §1 · p. 3
  • The proposed mesh-free KAN-PINN framework targets the biharmonic equation under fully clamped boundary conditions, combining a direct fourth-order residual, smooth activations, and automated multi-stage optimization.The paper identifies this setting as having numerical stiffness characteristic of high-order plate-bending problems.Source: §1 · p. 3

2 Mathematical foundations

The paper formulates a clamped-plate biharmonic problem on a bounded square domain and introduces its KAN-PINN approximation and physics-informed loss. The boundary data prescribe displacement and normal slope.

2 Mathematical foundations

  • The mathematical foundations section sets up the clamped thin-plate biharmonic problem and introduces the KAN-PINN approximation and corresponding physics-informed loss functional.Source: §2 · p. 4

2.1 The biharmonic equation

  • The model considers a thin elastic plate on the unit square under small-deflection, linear-elasticity, and Kirchhoff–Love plate-theory assumptions.Source: §2.1 · p. 4
  • Clamped boundary data prescribe displacement and normal slope along the boundary; for a homogeneous clamped plate, both data are zero.Source: §2.1 · p. 4

2.2 Weak formulation

The weak formulation uses an admissible function space for the clamped plate and expresses the variational problem through the Hessian or, equivalently, the Laplacian.

2.2 Weak formulation

  • Because the biharmonic operator is fourth order, the natural energy space for the clamped plate problem is a subspace of H2(Ω).The boundary conditions are understood in the sense of Sobolev traces.Source: §2.2 · pp. 4–5
  • Testing the governing equation with an admissible function and integrating by parts yields a variational formulation in terms of the Hessian.Source: §2.2 · p. 5
  • Under clamped boundary conditions, the bilinear form can equivalently be written in terms of the Laplacian.Source: §2.2 · p. 5

2.3 Existence and uniqueness of the solution

The weak formulation supports existence and uniqueness for suitable forcing, and the solution depends continuously on the forcing term. Nonhomogeneous boundary data can be handled through a lifting function.

2.3 Existence and uniqueness of the solution

  • For forcing f in H−2(Ω), the weak formulation establishes existence and uniqueness of the clamped-plate solution, with continuous dependence on the forcing term.Sufficiently regular loads may in particular be taken from L2(Ω).Source: §2.3 · p. 5
  • Classical C4(Ω) regularity for the PINN formulation requires forcing and boundary data to meet corresponding regularity and compatibility conditions, with corner singularities requiring care.Source: §2.3 · p. 6

2.4 Nonhomogeneous clamped boundary conditions

  • For nonhomogeneous boundary data, an appropriate lifting function incorporates the boundary values, after which the existence and uniqueness arguments still apply.Source: §2.4 · p. 6

2.5 Mesh-free KAN approximation

The KAN approximates the plate displacement as a mesh-free function, with trainable network parameters. The governing equation and boundary conditions are enforced at collocation points sampled from the domain.

2.5 Mesh-free KAN approximation

  • The KAN serves as a mesh-free function approximator for the unique solution of the boundary-value problem.Its trainable parameters θ define the approximation to the exact displacement field.Source: §2.5 · p. 6
  • The framework requires no predefined mesh, element connectivity, or nodal basis.The governing equation and boundary conditions are instead enforced at collocation points sampled directly from the domain.Source: §2.5 · p. 6
  • Interior and boundary collocation points are distinguished in the formulation.The passages identify these point sets but do not state their sampling counts or distributions.Source: §2.5 · p. 6

2.6 Physics-Informed Residual

The KAN-PINN imposes the biharmonic equation through a strong-form residual and enforces both clamped boundary conditions through separate residuals. Its parameters are trained by minimizing a composite loss over these residuals.

2.6 Physics-Informed Residual

  • The differential equation is imposed through a strong-form residual, whose derivatives are evaluated using automatic differentiation.The differentiable KAN structure supports this derivative evaluation.Source: §2.6 · p. 7

2.7 Boundary residuals

  • Two boundary residuals impose the clamped conditions: one for displacement and one for normal slope.The passages identify the two residuals but do not provide readable expressions for them.Source: §2.7 · p. 7

2.8 KAN-PINN loss functional

  • The KAN parameters are determined by minimizing a composite physics-informed loss of interior PDE and clamped-boundary residuals.The loss is given in a discrete least-squares formulation.Source: §2.8 · p. 7

2.9 Continuous least-squares interpretation

The discrete loss can be viewed as a quadrature approximation to a continuous residual functional. The KAN-PINN seeks a function that satisfies the interior equation and clamped boundary conditions simultaneously.

2.9 Continuous least-squares interpretation

  • The discrete loss is interpreted as a quadrature approximation to a continuous residual functional.The passage introduces the continuous functional as an alternative interpretation of the discrete loss.Source: §2.9 · p. 8
  • The exact solution has J(u) = 0 when all residuals vanish.Source: §2.9 · p. 8
  • The KAN-PINN seeks a function in its hypothesis space that simultaneously satisfies the interior equation and clamped boundary conditions.Source: §2.9 · p. 8

3 Neural network architecture

The mesh-free KAN-PINN represents plate deflection with learnable functions on network edges, using tanh between layers to retain high-order differentiability. The described architecture has four layers and 7,801 trainable parameters.

3 Neural network architecture

  • The mesh-free KAN-PINN represents transverse deflection globally, placing learnable one-dimensional functions on network edges instead of fixed node activations.Source: §3 · p. 8

3.1 Kolmogorov–Arnold network layer formulation

  • KAN edges combine a linear transformation with basis functions, and the layer formulation identifies a learnable bias, linear weight, and Gaussian-bump coefficients.The network also uses a fixed grid with dynamic refinement through knot insertion and grid extension.Source: §3.1 · p. 8, Figure 1 · p. 9

3.2 Approximation guarantees and neural scaling behavior

  • For cubic splines, the stated theoretical scaling exponent is α = 4 under the described smooth-layer representation assumption.Source: §3.2 · p. 9

3.3 Activation function justification

  • Tanh is used between layers because the biharmonic operator differentiates the network output four times and requires continuous high-order derivatives.The passage notes that ReLU's second derivative is zero almost everywhere, making it unsuitable for this residual.Source: §3.3 · p. 9

3.4 Network topology and parameterization

  • The topology maps spatial coordinates to scalar deflection through four layers with hidden width 20, totaling 7,801 trainable parameters.The architecture is 2 → 20 → 20 → 20 → 1.Source: §3.4 · p. 10, Table 1 · p. 10

4 Optimization strategy and training procedure

Fourth-order PINNs face severe numerical stiffness, so the framework combines a hybrid optimizer with automated search. On a manufactured unit-square benchmark, training reached a relative L2 error of 1.593×10^-5 and loss of 3.065×10^-6.

4.1 Hybrid first- and second-order optimization schedule

  • The hybrid schedule starts with AdamW exploration and switches after plateauing to restarted L-BFGS for precision convergence.The pipeline targets stiffness in fourth-order PINNs, where parameter-update noise can make raw-residual optimization stall or oscillate.Source: §4.1 · p. 10, §4 · p. 10

4.2 Automated train-change-improve exploration loop

  • The automated 24/7 search repeats training and applies structural or hyperparameter adjustments when optimization plateaus.Across 124 attempts, the search achieved 19 successive improvements and selected boundary penalty weights λdir = 480 and λneu = 30.Source: §4.2 · pp. 10–11

4.3 Manufactured solution and convergence monitoring

  • The manufactured benchmark monitors composite loss and ground-truth relative L2 error against an analytical target and source term.The source term is obtained by applying the biharmonic operator analytically to the target field.Source: §4.3 · p. 11
  • During training, composite loss and ground-truth relative L2 error were monitored; the error was measured against uexact, reaching 1.593×10−5 (0.00159%), while training loss reached 3.065×10−6.The benchmark uses the manufactured-solution technique on the unit square with an analytical target field.Source: §4.3 · p. 11

5 Numerical results and error analysis

The optimal mesh-free KAN-PINN closely approximates the manufactured unit-square solution, with low pointwise error and residuals across the domain. The results also track the hybrid optimizer’s convergence and the automated search’s successive error improvements.

5.1 Quantitative accuracy metrics

  • The optimal model uses 7,801 trainable parameters to approximate the manufactured biharmonic solution.Table 2 is identified as the summary of quantitative performance metrics, but its values are not supplied in these passages.Source: §5.1 · p. 11, Table 2 · p. 12

5.2 Field approximations, pointwise errors, and residual distributions

  • Maximum pointwise discrepancies remain on the order of 10^-5, while the interior residual distribution shows uniform adherence to the biharmonic equation.Figure 2 displays predicted deflection, absolute pointwise error, and absolute biharmonic residual across the domain.Source: §5.2 · p. 12, Figure 2 · p. 12
  • The predicted deflection qualitatively agrees with the manufactured sinusoidal solution, including its smooth symmetric peak and homogeneous clamped boundaries.The exact solution is sin(πx) sin(πy) on the unit square, with a peak of 1.0 at (0.5, 0.5) and u = 0 on all four edges.Source: §5.2 · pp. 12–13, Figure 3 · p. 13

5.3 Training and validation loss dynamics

  • Training and validation losses track closely over 2,000 epochs, with a sharp order-of-magnitude loss drop during the restarted L-BFGS phase.The loss history follows an initial steady descent under AdamW before the late L-BFGS polish.Source: §5.3 · p. 13, Figure 4 · p. 14

5.4 Error reduction via automated exploration

  • 0.00159% final relative L2 error is reached across 19 automated improvement steps, with restarted L-BFGS and a strong Dirichlet penalty identified as key convergence levers.The search trajectory is plotted on a log scale; the accompanying analysis reports negligible gains from larger networks, float64, or denser collocation sampling.Source: §5.4 · p. 14, Figure 5 · p. 15

6 Conclusion

The conclusion presents a mesh-free KAN-PINN framework for biharmonic boundary-value problems, combining edge-based learnable functions, smooth activations, a direct residual, and hybrid optimization. On the manufactured benchmark, the compact model attains a relative L2 error of 1.593 × 10^-5; future work targets more complex geometries and physical systems.

6 Conclusion

  • The framework combines edge-based radial-basis-function parameterization, smooth hyperbolic tangent activations, a direct normalized residual, and hybrid AdamW-to-L-BFGS optimization for fully clamped biharmonic problems.The authors report that this combination surmounted the numerical stiffness associated with the boundary conditions.Source: §6 · p. 15
  • 1.593 × 10^-5 relative L2 error and 3.065 × 10^-6 training loss are attained on the manufactured benchmark with 7,801 trainable parameters.The conclusion attributes systematic optimization of the restarted L-BFGS schedule and Dirichlet penalty calibration to an automated hyperparameter exploration loop.Source: §6 · p. 15
  • Future extensions are planned for irregular geometries, three-dimensional plate and shell structures, coupled multiphysics, and transient vibration phenomena.Source: §6 · p. 15

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