A differentiable Boussinesq framework for forward and inverse three-dimensional elastic contact

Finite element and boundary element methods (BEM) are reliable for forward elastic contact analysis, but inverse identification of material or loading parameters usually requires repeated forward solves or an external optimization loop. This work develops BEINN, a Boussinesq-equation-informed neural network for three-dimensional elastic half-space contact. A fixed differentiable Boussinesq layer maps pressure to surface displacement through zero-padded FFT convolution, so the network predicts only pressure while displacement and gap follow by construction. The forward loss combines Fischer–Burmeister complementarity with complementary-energy regularization and requires no labeled pressure data. For Hertz contact, BEINN matches the same-discretization BEM solution on a 128 × 128 grid with a pressure relative L² error of 0.003% and a load error of 0.0014%. The BEM reference is independently verified against the analytical solution by grid refinement, and the same hyperparameters remain accurate across three orders of magnitude in modulus. Fourier-MLP and U-Net backbones are used for smooth and rough contacts, respectively. BEINN also resolves deterministic rough-surface pressure fields, recovers modulus from sparse noisy observations, and is independently validated using analytical Hertz pressure data.