A Unified Phase-Field Fourier Neural Network Framework for Topology Optimization
We propose Alternating Phase-Field Fourier Neural Networks (APF-FNNs) as a unified and physics-based framework for topology optimization. The approach decouples the design problem by representing the state, adjoint, and topology fields with three separate Fourier neural networks, which are trained via a stable collaborative alternating scheme applicable to both self-adjoint and non-self-adjoint problems. To obtain well-resolved designs, the Ginzburg--Landau energy functional is embedded in the loss of the topology network as an intrinsic regularizer, naturally enforcing smooth and distinct interfaces between the two phases. Phase-field updates are driven by adjoint-based optimality conditions, and design sensitivities are evaluated efficiently using automatic differentiation, ensuring that the gradients correspond to exact total derivatives rather than naive partial derivatives. In contrast to classical phase-field methods, APF-FNNs exploit these physically consistent design gradients directly, avoiding pseudo-time gradient-flow solvers. By formulating physics-driven losses from variational principles or strong-form PDE residuals, the framework is broadly applicable to 2D and 3D benchmark problems, including compliance minimization, eigenvalue maximization, and Stokes/Navier--Stokes flow optimization. Across these examples, APF-FNNs consistently yield competitive performance and well-resolved topologies, establishing a versatile and scalable foundation for physics-driven computational design.