A physics-encoded Fourier neural operator approach for surrogate modeling of divergence-free stress fields in solids

TL;DR

This work develops a physics-encoded Fourier neural operator (PeFNO) to surrogate-model divergence-free stress fields in solids by embedding a stress-potential representation directly into the operator architecture. Unlike physics-informed approaches that constrain the loss, PeFNO enforces equilibrium at the output via $\mathrm{div}\,\mathbf{P}=\mathrm{div}\,\mathrm{curl}\,\tilde{\mathbf{A}}=\mathbf{0}$, yielding more accurate and robust satisfaction of mechanical equilibrium. The method is demonstrated on a polycrystalline unit cell under uniaxial extension, showing superior accuracy in stress fields and their divergence compared with physics-guided and physics-informed FNOs, particularly near grain boundaries. The work integrates Helmholtz-type decompositions for tensor fields, Fourier-space discretizations, and a mean-fluctuation framework to efficiently encode divergence-free constraints, offering a pathway toward data-efficient, physics-consistent surrogate modeling in solid mechanics. Future directions include incorporating deformation gradients into the inputs and applying gauge-based constraints (e.g., Coulomb gauge) to further improve invertibility and fidelity of the stress-potentials.

Abstract

The purpose of the current work is the development of a so-called physics-encoded Fourier neural operator (PeFNO) for surrogate modeling of the quasi-static equilibrium stress field in solids. Rather than accounting for constraints from physics in the loss function as done in the (now standard) physics-informed approach, the physics-encoded approach incorporates or "encodes" such constraints directly into the network or operator architecture. As a result, in contrast to the physics-informed approach in which only training is physically constrained, both training and output are physically constrained in the physics-encoded approach. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a "proof-of-concept" example application of the proposed PeFNO, a heterogeneous polycrystalline material consisting of isotropic elastic grains subject to uniaxial extension is considered. Stress field data for training are obtained from the numerical solution of a corresponding boundary-value problem for quasi-static mechanical equilibrium. This data is also employed to train an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO) for comparison. As confirmed by this comparison and as expected on the basis of their differences, the output of the trained PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the trained PgFNO or the trained PiFNO.

Figures (10)

• Figure 1: Schematics of physics-constrained NAs. In all cases, input (e.g., material properties, boundary conditions, or the deformation history) is transferred by the input layer (violet) to the hidden layers (green; only one is shown for simplicity). These in turn transform the input into $\mathbf{P}^{\textrm{out}}$ via an output layer (light red). See text for more details and discussion.
• Figure 2: Example microstructure in 2D unit cell $U$ ($x_{1}$ horizontal, $x_{2}$ vertical).
• Figure 3: Stress field data in $U$ for uniaxial extension of the microstructure in \ref{['fig:ProMatDat']} in the $x_{2}$ direction. All values in MPa. Note the differences in scale.
• Figure 4: Example components of $\mathbf{W}^{\mathrm{dat}}$ [1/$l$] for data in \ref{['fig:DatStsComCar']}.
• Figure 5: Comparison of $P_{\!22}^{\mathrm{dat}}$ and trained FNO results for $P_{\!22}^{\mathrm{out}}$ in $U$. All values in MPa.