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Neural networks have been applied within the Wave Digital Filter (WDF) framework as data-driven models for nonlinear multi-port circuit elements. Conventionally, these models are trained on wave variables obtained by sampling the current-voltage characteristic of the considered nonlinear element before being incorporated into the circuit WDF implementation. However, isolating multi-port elements for this process can be challenging, as their nonlinear behavior often depends on dynamic effects that emerge from interactions with the surrounding circuit. In this paper, we propose a novel approach for training neural models of nonlinear multi-port elements directly within a circuit’s Wave Digital (WD) discretetime implementation, relying solely on circuit input-output voltage measurements. Exploiting the differentiability of WD simulations, we embed the neural network into the simulation process and optimize its parameters using gradient-based methods by minimizing a loss function defined over the circuit output voltage. Experimental results demonstrate the effectiveness of the proposed approach in accurately capturing the nonlinear circuit behavior, while preserving the interpretability and modularity of WDFs.

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