Standard Physics-informed neural network (PINN) suffers from significantly degraded convergence and accuracy when dealing with complex geometries, heterogeneous materials, and discontinuous fields. Level-set-augmented PINN (LSA-PINN), which incorporates level-set functions as extra input features to encode prior geometric information near material interfaces, has demonstrated promising performance on benchmark problems with analytically representable two-material interfaces. However, extending these methods to practical problems with complex geometries remains challenging due to the lack of effective geometric representations for multi-material domains. We propose partition-wise R-function-based approximate distance functions (ADFs), in which a dedicated ADF is constructed for each subdomain to establish differentiable geometric representations of partitioned multi-material domains. Furthermore, an energy functional formulation is developed to eliminate both the loss terms involving second-order derivatives of the ADFs and the need to explicitly enforce interface conditions, thereby ensuring better optimization stability. Based on the proposed partition-wise ADFs, we develop an ADFs-augmented Deep Ritz Method (DRM), where an energy functional incorporating ADFs as additional input features is formulated for solving multi-material problems. The prominent strength of our method lies in its capability to handle complex geometries across multi-material regions without requiring any interface-specific penalty weightings. The proposed method is validated by comparison with the finite element method(FEM) and DRM enhanced with scalable tanh and Fourier features. Numerical results demonstrate that the proposed neural solver produces physics-conforming solutions with significantly improved accuracy.
This work introduces a variationally consistent, fully coupled dynamical framework for multi-objective topology optimization using the level-set method, shifting the formulation from a static state-selection problem to a continuous path-evolution problem. Conventional weighted-sum approaches rely on static scalarization and therefore maintain a fixed scalarization direction throughout the optimization process. To introduce an additional degree of freedom for the evolution of this direction, the level-set function and scalarization parameters are cast as coupled dynamical variables, tracing a continuous solution trajectory in a joint geometry-parameter space. Based on Hamilton’s principle augmented by non-conservative contributions, action functionals are formulated in both the objective and scalarization parameter spaces, yielding a coupled system comprising a second-order hyperbolic PDE and a set of ODEs. To satisfy the convex-combination constraint intrinsically, a nonlinear stick-breaking parameterization maps an open hypercube onto the open simplex, thereby satisfying the positivity and normalization constraints by construction for admissible parameter values. The resulting formulation delivers a mathematically consistent evolution law for the parameters, while the classical weighted-sum method is recovered analytically as an asymptotic limiting case. Furthermore, a variational energy analysis derives trajectory-dependent feasibility thresholds for the damping coefficients, leading to an analytically defined, self-regulating mechanism that adaptively suppresses spurious oscillations. Combined with a Courant–Friedrichs–Lewy-type condition, the framework incorporates automated control of the evolution step size. Finally, five numerical examples investigate the influence of the system parameters and demonstrate the stable evolution of the coupled solution trajectories within the considered linear-elastic problems.
