Understanding how spatial architecture controls transient thermal response is essential for designing functionally graded materials (FGMs) in aerospace, biomedical, and energy systems. In particular, when material gradation varies independently in both radial and circumferential directions—as in many hollow structures—the governing equations become highly intricate under transient conditions. This complexity is further exacerbated by circumferentially varying convective coefficients and spatiotemporal ambient-temperature fluctuations, which are rarely addressed in analytical FGM studies. To address this challenge, we present a unified analytical–machine-learning framework for predicting transient thermal behavior in asymmetrically graded hollow cylinders. The classical Fourier–Sturm–Liouville expansion is extended to orthotropic FGMs with circumferentially varying Robin conditions, inducing Toeplitz-coupled modal constraints. The resulting system is solved exactly using a projector–pseudoinverse formulation, yielding closed-form eigenfunction series solutions. Five graded architectures—power-law, exponential, sigmoid, and mixture—are analysed, revealing heat-transfer characteristics ranging from uniform heating to diffusive thermal buffering. Applications are demonstrated in biological tissue ablation and nuclear fuel-cladding design. Finally, analytical and numerical solutions are used as high-fidelity training data to improve operator learning and prediction efficiency in the multi-functional, gradient-enhanced physics-informed gPI-DeepONet. Compared with DeepONet, errors decrease by approximately , , and for the gPI-DeepONet, hybrid-FEA, and hybrid-analytical variants, respectively. • Introduces a novel Fourier-Sturm-Liouville formulation for heat conduction in orthotropic FGMs with Toeplitz coupling under angularly varying Robin conditions. • Develops a hybrid analytical-operator framework for transient heat transfer in asymmetric FGMs. • Demonstrates the use of high-fidelity analytical /numerical solutions to train operators for unseen boundary functions. • Reveals distinct thermal regimes from uniform heating to diffusive buffering across graded designs. • Achieves 95-98.3% lower L 2 error using gradient-enhanced gPI-DeepONet and hybrid models compared with DeepONet.
Das et al. (Sun,) studied this question.