In the domain of computational science and engineering, large, sparse linear systems are prevalent. A commonly embraced strategy for solving these systems is the use of preconditioned Krylov subspace iterative methods. For symmetric positive definite (SPD) matrices, the preconditioned conjugate gradient (PCG) solver is frequently employed. Nonetheless, designing effective preconditioners for the PCG solver requires specialized domain knowledge and meticulous matrix manipulations. Recently, Graph Neural Networks have been applied either to directly infer preconditioners from data [1, 2] or to leverage existing algebraic strategies for learning the target preconditioner [3]. In this work, we present a Graded Graph Spiking Convolutional Neural Network (GGSCN) that is embedded in the PCG solver and builds on classical preconditioners. By learning highly efficient preconditioners, the GGSCN reduces the number of iteration steps needed by the PCG solver. Unlike binary‑spiking neurons, the graded spiking units in the GGSCN propagate the membrane potential of active neurons, merging the benefits of conventional and spiking models for time‑dependent simulations such as Finite Element (FE) analysis [4]. Additionally, we propose the GGSCN to be trained by a weighted loss function comprising physics-based loss (the preconditioned residual equation evaluated at each iteration) and the spectral diameter loss. To fine-tune the coefficients of the loss functions, an agentic workflow is introduced by employing a Large Language Model (LLM) as a meta‑controller, which functions as a domain expert analyst. The LLM engages with the training loop periodically, reasons through its logic kernel and outputs optimised weights balancing the physics‑based residual and spectral‑diameter losses. The proposed approach achieves the research goal of substantially reducing PCG iterations and the system matrix’s condition number. Finally, we validate the method on dynamic viscoplastic impact simulations, benchmarking the GSCNN preconditioner against classical counterparts to demonstrate its efficiency and generalization for both linear and nonlinear FE models.