LT-PINN: Lagrangian topology-conscious physics-informed neural network for boundary-focused engineering optimization

Physics-informed neural networks (PINNs) have emerged as a powerful meshless tool for topology optimization, capable of simultaneously determining optimal topologies and physical solutions. However, conventional PINNs rely on density-based topology descriptions, which necessitate manual interpolation and limit their applicability on precise topology boundary and its normal reconstruction. To address this, we propose Lagrangian topology-conscious PINNs (LT-PINNs), a novel framework for boundary-focused engineering optimization. By parameterizing the control variables of topology boundary curves as learnable parameters, LT-PINNs eliminate the need for manual interpolation and enable precise boundary determination. We further introduce specialized boundary condition loss function and topology loss function to ensure sharp and accurate boundary representations, even for intricate topologies. The accuracy and robustness of LT-PINNs are validated via two types of partial differential equations (PDEs), including elastic equation with Dirichlet boundary conditions and Laplace's equation with Neumann boundary conditions. To demonstrate its broad applicability, we also implemented LT-PINNs on several primitive topologies and benchmarked its performance. The effectiveness of LT-PINNs is finally verified on more complex time-dependent and time-independent flow problems without relying on measurement data, and showcase their engineering application potential in flow velocity rearrangement, transforming a uniform upstream velocity into a sine-shaped downstream profile. The results demonstrate (1) LT-PINNs achieve substantial reductions in relative L₂ errors compared with the state-of-art density topology-oriented PINNs (DT-PINNs), (2) LT-PINNs can handle arbitrary boundary conditions, making them suitable for a wide range of PDEs, and (3) LT-PINNs can infer clear topology boundaries without manual interpolation, especially for complex topologies.