Mass weighting algorithm optimizes Fourier-based physics-informed neural network in adhesive contact mechanics

Abstract: Physics-informed neural networks (PINNs) for elastic contact mechanics suffer from a spectral stiffness imbalance,that is, the elastic kernel grows linearly with wave number, causing short-wavelength modes to dominate gradient updates and stall convergence of the macroscopic deformation. We introduce a spectral preconditioning strategy that reweights displacement gradients in Fourier space before back-propagation, amplifying low wavenumber components through a mass weighting (MW) function while suppressing sub-grid noise via a built-in low-pass filter. Applied to adhesive line contact problems, the mass weighted PINN reaches machine-zero residual loss within 400 Adam iterations for specified benchmark, whereas the reference benchmark stalls at three orders of magnitude higher loss. The converged displacement and contact stress fields agree quantitatively with Green's function molecular dynamics (GFMD) solutions for both smooth Hertz contact at pressures spanning tension to compression and rough surfaces with roughness covering several decades of wavelength. The method operates directly on a uniform real-space grid, requires no explicit Green's function integration or quadrature rules, and is formulated entirely in terms of minimising a scalar energy function. Extension to two-dimensional rough surfaces is direct, as both the Fourier elastic energy and the spectral preconditioner depend only on the wave-number magnitude.

• The paper introduces a mass-weighting algorithm that counteracts spectral stiffness in Fourier-based PINNs to ensure rapid and stable convergence.
• It employs a feedforward neural network with explicit Fourier feature mapping and spectral preconditioning to damp high-frequency noise during training.
• Numerical results on Hertzian and rough contact cases demonstrate that the method achieves normalized RMSEs as low as 10⁻⁶, aligning well with GFMD benchmarks.

Mass Weighting Fourier-Based PINN for Adhesive Contact Mechanics: An Expert Summary

Background and Motivation

The numerical solution of contact mechanics, specifically for adhesive interfaces featuring multi-scale surface roughness, presents substantial challenges. Traditional computational techniques such as FEM and BEM become increasingly intractable for highly multiscale or large systems due to meshing or all-to-all coupling overheads. Recently, PINNs have shown promise for physics-constrained modeling in contact mechanics, but their efficiency is impeded by spectral stiffness imbalances: short-wavelength (high-$q$) Fourier modes dominate gradient flows due to the linear growth of the elastic kernel, stalling convergence of long-wavelength (macroscopic) deformations essential for correct physical solutions. The paper "Mass weighting algorithm optimizes Fourier-based physics-informed neural network in adhesive contact mechanics" (2607.04288) introduces a spectral preconditioning strategy—mass weighting (MW)—that directly addresses this limitation within energy-minimizing Fourier-based PINNs.

Methodological Framework

The study formulates the adhesive line-contact problem as an unconstrained energy minimization, parameterizing the displacement field $u(x)$ using a feedforward neural network with explicit Fourier feature mapping. The total potential energy $\Pi[u]$ incorporates:

• Elastic energy evaluated spectrally via the $qE^*/2$ kernel,
• External work, and
• Adhesive/interface interaction energy captured by a Morse potential.

During training, gradients with respect to the displacement field are evaluated, Fourier-transformed, and then reweighted by a mass weighting function $w(q)$ that counteracts the spectral imbalance introduced by the elastic kernel. An additional spectral low-pass filter $f(q)$ suppresses sub-grid noise. The combined preconditioner gain $G(q) = w(q)f(q)$ is then inverse-transformed to real space, ensuring that gradient flows signal macroscopic modes sufficiently for stable convergence without manual hyperparameter tuning. This approach is implemented within a standard Adam optimization framework.

Numerical Results and Quantitative Performance

The algorithm is validated on two canonical cases:

• Classical Hertz (smooth parabolic) line contact across a spectrum of external pressures,
• Rough surface contact with a well-specified power spectral density (PSD) and Morse-type adhesion.

Convergence and Stability:

Spectral preconditioning with MW is demonstrated to be not just beneficial, but necessary. Without MW, PINN training quickly stalls—residual loss plateaus at three orders of magnitude higher than the converged value, and the displacement/stress fields exhibit pronounced high-frequency noise, failing to capture physically correct contact patches. With MW activated, convergence is achieved rapidly (machine-zero loss within 400 iterations), and the amplitude of the highest-$q$ modes is robustly damped after an initial network warm-up.

Physical Accord versus Reference Methods:

Displacement and stress fields recovered by the MW-PINN are quantitatively consistent with GFMD solutions for both benchmark geometries:

• For Hertzian contacts, normalized RMSEs (NRMSE) for $u$ and $\sigma$ are in the range of $u(x)$0–$u(x)$1 and $u(x)$2–$u(x)$3, respectively.
• For rough surface contacts, NRMSEs remain below $u(x)$4 for displacement and $u(x)$5 for stress.

At low applied pressures (sparse contact), the PINN cleanly reproduces the isolated, highly localized stress peaks characteristic of multi-scale rough contacts; at higher pressures, the predicted transition to nearly continuous contact stress aligns closely with GFMD.

Theoretical and Practical Implications

Theoretical Advancement

• Spectral Preconditioning for Energy PINNs:

The MW scheme directly targets the spectral structure of the elastic operator within the PINN’s variational framework, providing a frequency-aware gradient flow that circumvents the convergence pathologies identified in the literature [Rahaman et al. (2019); Wang et al. (2021); He et al. (2026)]: namely, strong operator-induced gradient-flow stiffness not addressable by architectural or standard optimizer modifications alone.

• Physics-Ensemble Flexibility:

The preconditioner is formulated generically for any energy-based PINN framed in Fourier space, provided the dominant spectral contribution to the gradient is elastic in origin. This hints at direct extensibility to other classes of mechanical or interfacial PINN problems, abstraction beyond strictly adhesive or Morse-type potentials, and potential applicability to multi-physics settings.

Practical Utility

• Meshless and Green’s Function-Free Implementation:

By leveraging a uniform real-space grid and spectral evaluation, the approach circumvents the need for explicit Green’s function inversion or quadrature, reducing algorithmic overhead and facilitating large-scale simulation in scenarios where BEM or volumetric FEM would be computationally restrictive.

• Robustness and Automation:

The lack of requirement for hyperparameter tuning of the MW scheme or network initialization protocols means the method is well-poised for automation, including for high-throughput investigations of contact in engineering and tribology.

• Direct Path to High-Dimensional Surfaces:

The spectral structure underlying MW generalizes immediately to higher dimensions (e.g., 2D rough surfaces), making extension to technically relevant problems straightforward.

Future Directions

Potential avenues for advancing this line of work include:

• Extension to full 2D contact mechanics for realistic rough surfaces, possibly with non-Gaussian height statistics relevant for polymeric or biological materials.
• Incorporation of more complex, possibly rate-dependent or history-dependent interface laws (viscoelasticity, rate-and-state friction).
• Exploration of the preconditioner’s efficacy in PINN models for other physics, including fracture, delamination, or coupled hydro-mechanics at interfaces.
• Investigation of adaptive or learning-based spectral preconditioners that can adjust to operator structure not known a priori.

Conclusion

This study decisively establishes the necessity and efficacy of Fourier-domain mass weighting as a spectral preconditioner for energy-minimizing PINNs in contact mechanics. By rectifying the spectral stiffness imbalance of the elastic kernel, the mass weighting scheme facilitates rapid, robust, and physically accurate convergence for both classical and multiscale rough adhesive contacts relative to GFMD reference solutions. The methodology is extensible, non-intrusive, and well-aligned with the spectral properties intrinsic to many physics-informed learning problems in continuum mechanics and interface science.

Reference:

"Mass weighting algorithm optimizes Fourier-based physics-informed neural network in adhesive contact mechanics" (2607.04288)