S-Crescendo: A Nested Transformer Weaving Framework for Scalable Nonlinear System in S-Domain Representation

TL;DR

S-Crescendo presents a physics-informed, nested Transformer framework that weaves S-domain priors with neural operators to predict time-domain responses of high-order nonlinear RC networks. By leveraging partial fraction decomposition, the method decouples modal responses and uses a baseline first-order predictor plus residual corrections, achieving near-$$R^2$$ accuracy (up to $R^2\approx 0.99$) while reducing computational complexity from $\mathcal{O}(n^3)$ to $\mathcal{O}(n^2)$ (and $\mathcal{O}(n)$ per output). The approach delivers substantial speedups over HSPICE (up to ~18x) and generalizes from orders 1–3 to higher orders (4–9) with strong predictive performance, enabling scalable, physics-aware nonlinear modeling in VLSI design. Its modular architecture and MOR-compatible formulation offer a practical path toward efficient simulation of high-dimensional nonlinear systems and potential extension to general nonlinear–linear hybrid domains.

Abstract

Simulation of high-order nonlinear system requires extensive computational resources, especially in modern VLSI backend design where bifurcation-induced instability and chaos-like transient behaviors pose challenges. We present S-Crescendo - a nested transformer weaving framework that synergizes S-domain with neural operators for scalable time-domain prediction in high-order nonlinear networks, alleviating the computational bottlenecks of conventional solvers via Newton-Raphson method. By leveraging the partial-fraction decomposition of an n-th order transfer function into first-order modal terms with repeated poles and residues, our method bypasses the conventional Jacobian matrix-based iterations and efficiently reduces computational complexity from cubic $O(n^3)$ to linear $O(n)$.The proposed architecture seamlessly integrates an S-domain encoder with an attention-based correction operator to simultaneously isolate dominant response and adaptively capture higher-order non-linearities. Validated on order-1 to order-10 networks, our method achieves up to 0.99 test-set ($R^2$) accuracy against HSPICE golden waveforms and accelerates simulation by up to 18(X), providing a scalable, physics-aware framework for high-dimensional nonlinear modeling.

Figures (4)

• Figure 1: Given a known input signal, the task is to predict the nonlinear system’s output before it feeds into the linear system. This intermediate signal, marked in red, is unknown and serves as the prediction target of our model.
• Figure 2: Model Overview. The model first constructs a baseline prediction by summing the first-order responses of pole-residue pairs. For each mode $i$, the residual module $e_{i}$ is trained using the current and previous poles, residues, and time information to correct the accumulated error. Residuals are added iteratively to refine the final prediction.
• Figure 3: (a) Model performance on single-pole transfer functions, showing minimal overfitting and excellent test-set accuracy. (b) Effectiveness of our recursive error-correction module on a three-pole example.
• Figure 4: Prediction vs. true response for transfer‐function orders 4 to 9.