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

Technical Summary: Mass Weighting Algorithm for Fourier-Based PINNs in Adhesive Contact Mechanics

Problem Statement
Physics-informed neural networks (PINNs) applied to elastic contact mechanics face a fundamental challenge known as spectral stiffness imbalance. In elastic half-space problems, the elastic kernel in Fourier space scales linearly with the wave number ($q$). Consequently, short-wavelength (high-$q$) deformation modes incur significantly higher elastic energy costs than long-wavelength modes. When training standard PINNs via gradient descent, this imbalance causes high-frequency modes to dominate the gradient updates, leading to the stalling of convergence for macroscopic deformation fields. This issue is exacerbated in adhesive contact problems where short-range interactions introduce additional high-frequency content. Standard energy-minimizing PINNs often fail to converge to machine-zero residuals or produce unphysical, oscillatory stress fields dominated by high-frequency noise.

Methodology
The authors propose a Fourier-based PINN framework augmented by a spectral preconditioning strategy called Mass Weighting (MW). The methodology involves the following key components:

1. Energy-Minimizing Formulation: Unlike traditional PINNs that balance PDE residuals with boundary condition penalties, this approach formulates the learning task as the minimization of a scalar total potential energy function ($\Pi$). The total energy comprises elastic strain energy ($U_{el}$), external work ($U_{ext}$), and interfacial interaction energy ($U_{int}$) modeled by a Morse potential.
2. Fourier Feature Mapping: The input spatial coordinates are encoded using Fourier features to enable the network to resolve sharp displacement gradients near contact edges.
3. Spectral Preconditioning (Mass Weighting): Before back-propagation, the gradient of the loss function with respect to the displacement field is transformed into Fourier space. It is then reweighted by a mass weighting function $w(q)$ and a built-in low-pass filter $f(q)$:
   • Mass Weighting: $w(q) = \left( \frac{E^* q_{max}/2}{\sqrt{k_0^2 + (qE^*/2)^2}} \right)^\alpha$. This function amplifies low-wavenumber components (macroscopic modes) while suppressing high-wavenumber contributions, effectively rebalancing the gradient flow.
   • Low-Pass Filter: $f(q) = \frac{1}{1 + (q/q_c)^\beta}$. This suppresses sub-grid noise that cannot be resolved by the discretization.
4. Training Protocol: The preconditioner is activated after a "warm-up" phase of 100–200 iterations using plain gradients. The network is optimized using the Adam optimizer. The contact stress is recovered via inverse Fourier transform of the spectral stress-displacement relation, avoiding grid-sensitive numerical differentiation.

Key Contributions

• Identification of Spectral Stiffness: The paper explicitly attributes the convergence failure of energy-based PINNs in contact mechanics to the spectral stiffness imbalance inherent in the elastic operator, rather than just general optimization difficulties.
• Mass Weighting Preconditioner: The introduction of a specific spectral preconditioner that operates in Fourier space to reweight displacement gradients. This allows the network to learn macroscopic deformations without being overwhelmed by high-frequency noise.
• Direct Energy Minimization: The formulation relies entirely on minimizing a scalar energy function without explicit Green's function integration or quadrature rules, yet achieves accuracy comparable to methods that do.
• Extension to Adhesive Rough Surfaces: The method is successfully applied to both smooth Hertzian contacts and fractal rough surfaces with adhesive interactions (Morse potential), covering regimes from tension to compression.

Results
The proposed method was validated against Green's Function Molecular Dynamics (GFMD) solutions, a high-fidelity benchmark for contact mechanics.

• Convergence Speed: For a benchmark Hertz contact problem, the mass-weighted PINN reached machine-zero residual loss within 400 Adam iterations. In contrast, the unconditioned (reference) training stalled at a loss three orders of magnitude higher ($\sim 5 \times 10^{-4}$) and failed to converge further.
• Physical Accuracy:
  • Smooth Surfaces: The PINN predictions for displacement and contact stress were indistinguishable from GFMD solutions across tension, zero-load, and compression regimes.
  • Rough Surfaces: For fractal rough surfaces (Hurst exponent $H=0.5$), the method accurately captured the multi-scale stress structure, including sparse asperity-level contact patches at low pressures and continuous stress distributions at high pressures.
• Quantitative Error: The Normalized Root-Mean-Square Error (NRMSE) for displacement and stress remained below $10^{-3}$ (typically $10^{-4}$ to $10^{-5}$) across all test cases, confirming quantitative consistency with the GFMD reference.
• Stability: Monitoring the highest-wavenumber Fourier mode revealed that unconditioned training resulted in persistent large-amplitude oscillations, whereas the mass-weighted approach rapidly decayed these modes to zero, ensuring a stable, oscillation-free stress profile.

Significance and Claims
The authors claim that spectral preconditioning is not merely helpful but necessary for the stable convergence of energy-minimizing PINNs in contact mechanics. Without it, the network fails to learn the physically correct solution due to gradient imbalance. The proposed Mass Weighting algorithm enables the PINN to:

1. Solve adhesive contact problems on uniform real-space grids without explicit Green's function integration.
2. Achieve machine-zero residuals and smooth, physically correct stress fields on fine grids (up to $N_x = 2048$).
3. Operate robustly without manual tuning of preconditioner parameters beyond initial setup.

The paper concludes that while currently limited to one-dimensional line-contact geometries and Morse potentials, the Fourier-based formulation and the spectral preconditioner are directly generalizable to two-dimensional rough surfaces and other smooth interaction potentials, offering a scalable alternative to traditional boundary element methods.