Variational energy-embedded physics-informed neural network framework with analytic lifting for transient radial heat conduction

This study develops a variational energy-embedded physics-informed neural network (PINN) framework for transient radial heat conduction in wellbore thermodynamics. The heat equation is recast on a two-dimensional Cartesian annulus and solved through a semi-discrete weak energy minimization, so that only first-order spatial derivatives are required while the axisymmetric radial response is preserved. To resolve the steep thermal layer near the wellbore, an analytic lifting function based on the short-time diffusion kernel is combined with a polynomial boundary envelope to enforce the essential Dirichlet conditions directly in the trial space. This lifting term supplies the dominant boundary-layer prior for the reference annular benchmark, while the adaptive Fourier-embedded residual (AFR) network represents a trainable correction field for residual spectral variations and extended settings. The same weak-form setting is further extended to temperature-dependent diffusivity through a Picard-linearized frozen-coefficient update, avoiding a switch to pointwise strong-form residual minimization. Numerical results on an annular wellbore benchmark show improved near-wellbore accuracy relative to a reference strong-form PINN baseline, while maintaining small angular variance on the Cartesian annulus. A non-axisymmetric annular diagnostic further examines whether the Cartesian AFR trial space can represent angular temperature variations without explicit polar-periodic boundary constraints. The predicted thermal fields are also post-processed into thermoelastic displacement and stress responses and compared with GEOS calculations performed on a three-dimensional hexahedral quarter-cylinder mesh. These comparisons indicate that the proposed weak-form AFR framework is a useful neural trial-space construction for the benchmarked class of circular annular wellbore heat-conduction and thermoelastic post-processing problems. The evaluated scope is defined by the circular annular geometry, the tested diffusivity ranges, and the present thermoelastic post-processing assumptions.