S-Crescendo: A Nested Transformer Weaving Framework for Scalable Nonlinear System in S-Domain Representation
Authors: Junlang Huang
Chen Hao
Li Luo
Yong Cai
Lexin Zhang
Tianhao Ma
Yitian Zhang
Zhong Guan
Venue: NeurIPS 2025 poster ()
Keywords: nonlinear systems
VLSI backend design
transformer
transfer function
Primary area: machine_learning_for_sciences
TL;DR: Our method decomposes n-th order transfer functions into first-order modal terms, bypasses conventional matrix-based iterations and efficiently reduces computational complexity from O(n^3) to linear O(n).

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\(\times\), providing a scalable, physics-aware framework for high-dimensional nonlinear modeling.

S-Crescendo: A Nested Transformer Weaving Framework for Scalable Nonlinear System in S-Domain Representation
Junlang Huang", Hao Chen Li Luo" Yong Caili, Lexin Zhang', Tianhao Ma', Yitian Zhang', Zhong Guan 1s
Abstract
Simulation of high-order nonlinear system requires extensive computational re- sources, especially in modern VLSI backend design where bifurcation-induced instability and chaos-like transient behaviors pose challenges. We present S- Crescendo 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- Rapton-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 itera- tions and efficiently reduces computational complexity from cubic ((n²) linear (n)..Thn The proposed architecture seamlessly integrates an S-domain encoder with attention-based correction operator to simultaneously isolate dominant response and adaptively capture higher-order non-linearities. Validated on order- to order- 10 networks, method achieves up to 0.99 test-set R² accuracy against HSPICE golden waveforms and accelerates simulation by up to 18 x, providing scalable, physics-aware framework for high-dimensional nonlinear modeling.
Introduction
In recent years, deep learning technologies have made remarkable progress, with Transformer-based architectures demonstrating exceptional performance and significant advantages in fields such as natural language processing (NLP), computer vision, and time-series data modeling.[1] Transformer models, by effectively capturing complex relationships and long-range dependencies, offer novel perspective for data-driven modeling. This technological advancement has inspired researchers to explore its potential applications in traditional engineering domains, especially in complex physical modeling and signal prediction[2][3).
Nonlinear system identification remains a core challenge across many domains, particularly under high-order dynamics, non-stationarity, and limited observability. Classical methods from control thcory-such as Volterra series, Hammerstein-Wientt decompose system behavior into a linear core and a nonlinear correction [4, 5, 6, 7J. However, their scalability and generalization degrade in high- dith-mmmeniionll parameter spaces [8]. These limitations are especially pronounced in modern integrated circuit design, where nonlinearities emerge not only from active devices but also from parasitic effects in passive interconnects. A canonical example the "nonlinear driver linear RC load" configuration, illustrated in Figure 1. As technology scales, interconnect parasitics exhibit strong dynamic nonlinearities due to proximity effects, process variation, and material inhomogeneity [9], complicating accurate modeling. To address this, we propose a neural operator framework that integrates Laplace-domain physical priors with data-driven
models, and grey-box approaches-ttyically
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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adaptability. Demonstrated on RC current response tasks, this method extends naturally to systems governed by partial differential equations with nonlinear boundary conditions, offering scalable and physically consistent approach to complex system identification.
Manilaarr Syeten
What predet
Uneer
Sap
Figure I: Given a known input signal. the task is to predict the nonlinear system's output before feeds into the linear system. This intermediate signal, marked in red, is unknown and serves as the prediction target of our model.
In response, growing body of research has applied deep learning to nonlinear system identification. Approaches based on recurrent networks[10], attention mechanisms{ I], and neural operators [2] (e.g., DeepONet[ 13], FNO[14], and PINNs) have shown promise, particularly in low-dimensional settings or for PDE systems with known boundary conditions. Yet, these methods often require large datasets, lack robustness structural variation, and suffer from poor physical interpretability[I1]]16 More critically, few methods explicitly leverage the modal structure or transfer function representation intrinsic to many engineered systems. As result, they remain constrained by either computational inefficiency (due to iterative solvers like Newton-Raphson) or limited generalization to systems with unseen topologies or higher-order explored for general system identification, its application to signal-lins RC response modeling canonical high-order nonlinear problem booblem-remains remaiin largely unexplored. Most existing methods still fall under three classical paradigms: Current Source Models (CSMs) [20, 21], Voltage Response Models (VRMs) [22], and Direct Waveform Prediction [23]. Each faces practical limitations: CSMs lack waveform fidelity due to fixed capacitance abstraction [24], VRMs suffer from high cost and colver-induced errors [25, 26), and direct fitting fails under sharp transitions due to overshoot and undershoot distortion.
[
While deep learning has been
Against this backdrop, Transformer-based methods offer powerful new tool for nonlinear system identification in the S-domain. Transformer architectures excel at capturing complex, long-range dependencies and higher-order interactions, making them ideally suited to address the limitations of traditional RC-network models in time-domain response prediction [1].
Even nominally high-order interconnects, only a handful of "dominant poles" govern signal behavior. Standard model order reduction (MOR) selects these modes, typically fewer than ten by energy or gain thresholds, and collapses 5000th order RC network into a low-order surrogate with minimal frequency response error [27, 28). Going beyond linear reduction, we encode each reduced first-order term into a learned latent embedding and apply an attention-driven correction operator to capture nonlinear driver-load effects. On 4th to 10th order surrogates, our method attains R²2 up to 0.99 against HSPICE waveforms, evidencing superior accuracy. By leveraging partial-fraction decomposition, decouple topology from in the S-domain before summation, thus ensuring universality and structural agnosticism. We introduce physics awared S-domain neural operator that seamlessly integrates with MOR pipelines, delivering a scalable, accurate, and efficient solution for nonlinear RC simulation, electromagnetic analysis, and iignal-integrin evaluation without fixed driver load models.
foreccttinno each modal response independently
The proposed model adopts two-stage architecture: a base module first predicts individual first- order responses from partial fraction decomposition and aggregates them into a baseline output. A subsequent compensation network then iteratively refines this output by learning residual corrections across model orders. This overall architecture is illustrated in Figure 2.

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Ai
Fd Roult
Fredey
Ougger
RC
Figure 2: Model Overview. The model first constructs baseline prediction by summing the first-order responses of pole-residue pairs. For each mode the residual module e, 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.
2 Theoretical Background
2.1 Decomposition Theorem for High-Order RC Network Transfer Functions
Modeling nonlinear dynamical systems remains fundamental across many physical domains, where complex interactions often overlay intrinsic linear behaviors. In integrated circuit design, a prominent instance arises in interconnects, where nonlinear drivers interface with passive metal wires forming linear time-invariant (LTT) networks. The passive portion, shaped by parasitic resistances (Rt) and capacitances (Ct), governs signal delay, attenuation, and waveform distortion. The behavior of these RC networks is typically characterized in the Laplace domain via a transfer function H(s) that maps input signals output responses [27, 29, 4]. For network with independent energy-storue elements, the transfer function takes the form shown in Equation (1):
N(s) bo + bis S + 0m-
(()) D(s) On + 81s
am S
where the roots of the denominator (s)) are the system poles = = - all located on the negative real axis due to the passive nature of RC circuits.
By the fundamental theorem algebra and the Heine-Bou theorem, any strictly proper rational function with distinct poles admits unique partial fraction expansion[30] shown in Equation (2):
H(s)
>
m
T
N(s) D'(x)
This decomposition can be rigorously derived through two classical approaches. First, the method of undetermined coefficients constructs a linear system whose solution is guaranteed by the nonsingular- ity of the associated Vandermonde matrix when all poles are distinct. Second, the Cauchy residue theorem establishes the residue- based representation by integrating (s) around a closed contour enclosing all poles; analytic continuation then ensures the uniqueness of this expansion.
Further physical constraints arise from the realizability conditions of RC networks. All poles must lie strictly in the negative real domain to ensure overdamped and stable dynamics. While residues are typically real-valued, they may either positive or negative, reflecting modal interference effects in higher-order coupled systems. Despite potential non-monotonicity the modal level, the overall response remains physically consistent and interpretable, capturing the multi-timescale nature of signal propagation inherent to real interconnect behavior.

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2.2 Generalization to Repeated Poles
When pole p; of the transfer function has multiplicity k (with E, K4 m.), the partial -raiaan expansion naturally extends to form in Equation (3):
rig
H(s)
(3)
(s pi)
where each higher-order residue is given by Equation (4):
1 d
Fiy (k:- j)! j)! ds k-j [(s-
- p:) (s)]
(4)
In practice, however, exact repeated poles rare in on-chip RC networks due manufacturing tolerances and layout variations. Even when poles are nearly coincident, the corresponding higher- order residues riy tend to be small, and their time domain contributions 1 decay rapidly for P; < O. Consequently, one can safely ignore repeated-pole terms in most modeling tasks and rely on single-pole expansions to achieve high-fidelity simulations [31].
This comprehensive decomposition provides unified framework for both time-domain and frequency- domain analysis of arbitrary high-order RC interconnect networks.
2.3 Computational Complexity Analysis
Traditional SPICE-based transient simulation begins by formulating system of nonlinear equa- tions based on circuit devices and Kirchhoff's laws. Solving this system-typically via the New- ton-Raphson method-is computationally expensive. For an n-node RC network, each iteration involves Jacobian construction and LU decomposition, resulting in complexity of O Even with sparse solvers, fill-in effects lead an effective cost between (n2 and Onn)) over time steps and P ports [32, 33].
In contrast, method operates in the S-domain and eliminates matrix inversion. Computing the admittance or impedance to a single output node requires ((n) operations. Extending this to all n nodes yields a total complexity of Onn²). while avoiding iterative linear system solves.
3 Data Acquisition and Preprocessing
3.1 Simulation Environment Setup
The HSPICE simulation platform employing 40-nm CMOS PDK ensured process-compling device parameters (Vu, X. 4jea)), where a single-stage CMOS driver with IEEE 1481-2009-co commplaaan RC networks (parasitics extracted via Python) converted to an S-domain transfer function; subsequent partial fraction decomposition yielded first- order subsystems characterized by poles Pi and residues r; for neural operator-based time domain prediction.
modeled through state space representation and
3.2
Stimulation Configuration
The input voltage waveform was configured as an ideal step signal (0 to VDD transition). Transient simulations covered both the signal rise phases (0 to 20 ns) and steady state behavior.with time step resolution 10ps.
3.3 Feature Representation and Supervision
Each sample is defined by a tuple of conditioning inputs and supervised outputs. The conditioning inputs comprise a device type label, a sequence of transient time points t,...., tr., and a set of frequency domain features obtained via transfer function decomposition, encoded resi-ue pairs {pr, r-) Together, these inputs capture the structural, temporal, and modal characteristics of the circuit.
The supervised target the voltage response sequence (Vout(4),.... Vout (lT)) obtained from HSPICE simulation, which guides training via time-aligned regression.

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3.4 Feature Normalization
To harmonize heterogeneous input features and enhance model robustness, we apply the following transformations in single step shown in Equation (5):
V(() min(V)
t' log (1) Prr
(1))) max min(V)
(5)
dt
where (I)) is the original voltage at time i, min(V) and max(V) are its minimum and maximum over the waveform, è is a sampled transient time point, and f4; o; are the mean and standard deviation of 10) (t) across all samples. These normalizations place both features on comparable scales, mitigate the influence of outliers, and promote stable, efficient training.
3.5 Transfer Function-Basee RC Network Modeling
We propose a compact, system-level modeling framework for standard-cell-oe RC interconnects by decomposing the Laplace-domain transfer function shown in Equation (6):
Vour Vin
1
A
H(s)
i=1
where each decay rate p; 0 (inverse time constant) and residue A, R satisfies Σ, A; ensuring normalized unit-step response. This form captures the dominant exponential kernlls e -Ppt without explicit node-level modeling.
To enable neural-network learning and generalization across circuits of varying size, we encode each mode as pole-residue pair (p:: A,), normalize all p; and A, by max; Pt and max; |A,]. sort pairs by descending |A.]. This interpretable mode sequence accurately reconstructs high-order RC responses with linear complexity and full spectral fidelity.
Model Architecture Design
4.1 Baseline Module: First-Order Prediction
The baseline module predicts the nonlinear voltage response of a single-mode RC system, specified by pole-residue pair (p, r), at discrete time points (tx) Unlike ideal linear RC networks, the input waveform here first passes through nonlinear active components (e.g., CMOS drivers), making the overall system response analytically intractable. To address this, we employ a neural function approximator: (4x) (p, 4x) J-where Se is lightweight Transformer trained to capture the nonlinear mapping from modal and temporal inputs to voltage outputs[1].
The model consists of three encoder and three decoder layers, each composed of multi-head self- attention, ferward sublayers with GELU activation, and layer normalization(34] Positional encoding is included to preserve temporal structure. Input features (p, (&) are embedded and processed in parallel to predict (t+) at each time step. The network is trained end-to-ent using the Adam optimizer with weight decay, minimizing the mean squared c3ror[35][36] shown in Equation (7):
1 (v(tx) T E
MMSE
This architecture provides an accurate and generalizable first-order predictor that forms the foundation for modeling higher-order RC systems through residual correction.
4.2 Compensation Module: Residual Correction
The compensation module iteratively refines the baseline prediction by learning the residual error between successive model orders. For each order n and time point tąs the input feature vector is (n, Pai Fas Pn-1; Fx-1, tz) wwhere (Pn,7,) and (P=-1;;7=11) denote the pole-residue pairs for the nth and (n 1)th modes, respectively, The network outputs shown in Equation (8):
5

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(ta) (n,
(2,,Po Pni Pn-1:7n-1,
(8)
which represents the corrective residual to be added to the order-n prediction at time Êy-
The function 9r is implemented as lightweight Transformer with the same depth and hyperparam- as the baseline module. It is trained end-to-end using the AdamW optimizer to minimize the residual mean squared error, thereby progressively correcting and refining higher-order predictions.
4.3 Recursive Training Procedure
Let (p)) be the pole-residue pairs for the nithorrder model and (44)2=1 the sampled time points. Denote by fo the trained first-order predictor and by {E,,Jz=1 the sequence of learned residual modules up to order n 1. To train the nth residual module cØ.,, we first accumulate the baseline prediction by summing (ps, ra, fk) for 1, .,.... We then add all previously learned corrections çø, to form the current prediction and compute its discrepancy from the true output (4x). This residual error serves as the target cs.,, which is fit by minimizing the mean squared error over & = L...., ' using the AdamW optimizer. Repeating this process for each order yields cascade of lightweight modules that progressively refine the high-order RC response.
Algorithm Iterative Residual Correction Training
Require: Base predictor fo. residual module set {Es, f==1 sampled data (t&, Vout (tx))==1
Ensure: Trained residual modules {eo, )
1: Initialize cumulative baseline prediction: Vhase (f&) +- 0, Ak E [1,2]
2: for residual index j = to N do
3: Load current pole P, and residue Ty
Phase 1: Baseline prediction update
4: for time step k: = 1 to T do
5: Update baseline: Viase (t&) (onso(ta) fo fo
6: end for
Phase 2: Residual target computation
7: for time step k = 1 to I do Compute current prediction:
8:
(t*+) + Vhasse (tx) +
ea Pi
9:
Compute residual target:
(t(e) + Voue(ts) - (t(t
10:
end for
Phase 3: Module training
11:
Build training set: Dj = ((te,,((t)))1
12:
Minimize the loss:
T Σ T
min 65
(e.0 -rr(())))
1=1
13: Update Φy using gradient descent
14: end for
By repeating this procedure for n = 1,2...., we ensure each co.. learns to generalize the correction from order n- to n, yielding cascade of residual models that together approximate full high-order response with minimal degree of generalization. Each residual module es., depends only on the pole-residue pairs of two adjacent orders and the current time point, without requiring knowledge of the full network topology
Moreover, this recursive training scheme offers

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or total number of nodes. As result, the trained modules can be reasonably extended to refine predictions for moderately higher-order systems beyond those seen during training.
4.4 Inference Procedure
In the inference stage, we compute a single forward-Pp -pass estimate of the output waveform by first assembling baseline response and then applying all residual corrections in parallel. Specifically, for each time sample & shown Equation (9):
N
Vhase (tx)
=
Σ 1=1
folpi, ri, r.)
where p; is the ith pole (inverse time constant) of the transfer function, T; is the corresponding residue (modal weight), and [o(ps, re, (x) denotes the base predictor's output, typically rye -Pti,
All N residual modules Egy are then evaluated and summed in parallel shown in Equation (10):
Vv Vaase (tx)
Cs, (3
(10)
where (.)(.) is the jth trained residual correction module, and its inputs include the current and previous pole-residue pairs as well the time tg.
This procedure yields the final prediction Vn (+&) each ty with O(N) complexity, running 50 faster than commercial tools like HSPICE by requiring just a single pass through the base predictor and residual modules.
5 Experiment
5.1 Single-Pole Transfer Function: Training and Test Performance
We first evaluate the performance of our model on single-pole transfer functions. The model achieves near-perfect fit on the training data and strong generalization unseen single-pole functions.
1.1
1.1
1.0
1.0
prodDR-2000.22
pred
prac2:5-200.9997
8.7
6.7
real
8.0-
8.3
6.3-
0.0
6.0
log time
log time
(a) Single-Pole Transfer Function: Training vs. test performance. n²2 = 0.999 illustrates near-perfect fit the training data and strong generalization held-on single-pole examples.
This figure illustrates the step-by-step waveform correction process, from the raw prediction without any correction through the successive application of first, second, and third-orded error modules, culminating in final fit of
(b) Error-Correctie Conre-CCrrcctnn Model:
R² 0.9975
Figure 3: Model performance on single-pole transfer functions, showing minimal overfitting and excellent test-set accuracy. (b) Effectiveness our recursive error-correction module on a three-pole example.
5.2 Effectiveness of the Error-Correctia Model
To demonstrate the benefit of our recursive error-correction module, we evaluate on three-pole transfer function example shown in Equation (11):
7

---

3
A 8
((s)
Σ
(11)
where {pi} and {A:} are chosen such that the poles are well separated. Figure 3 shows the predicted and true step responses over time. We compute the coefficient of determination and obtain R² 0.9975, confirming that the ertor-correction stage significantly improves accuracy over the base model.
Generalization to Higher-Order Transfer Functions
Next, we evaluate the model's ability to generalize beyond training orders. We train exclusively on systems of order within and then evaluate on transfer functions of orders from 9. Table reports MSE and &2 on each higher-order test set. Despite never having seen orders above 3, the model retains strong predictive power, demonstrating effective extrapolation. extrapolation, additionally train error-correction model 15th-order systems and test it on 200th-order transfer functions; the full setup and per-case results are provided in the appendix. Across 20 held-out cases, the mean R reaches 0.983 (see Appendix).
probe more extreme
Order
8
R
0.986
0.997
0.995
0.990
0.979
0.964
Table 1: Generalization performance higher-order transfer functions (trained orders 3).
(a) Order 4
(b) Order 5
(c) Order 6
exbpedt
(d) Order 7
(e) Order 8
(f) Order 9
Figure 4: Prediction vs. true response transfer-function orders to 9.
5.4 Inference Time Comparison
Training cost. Each
module contains approximately 100K-160K parameters. All models were trained on a single NVIDIA RTX 4090 GPU. For reference, single-por systems typically train in under 20 minutes, while 10th-order systems converge in approximately 2 hours. Given the considerable acceleration our model achieves at inference time, this one-time training cost highly acceptable.
Dataset construction. Training data generated using standard HSPICE simulations. Each simula- tion typically completes in about 0.5 seconds. For our full training dataset of 5,000 samples, the

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end-to-end preprocessing pipeline (simulation, waveform extraction, and supervised data formatting) completes in under 2 hours. The process fully scriptable and parallelizable, and does not pose practical bottleneck.
Runtime comparison. Finally, compare the runtime of our S-Crescendo model against HSPICE on a 10 ns transient simulation sampled at 1,000 time steps. HSPICE runtimes for orders through 10 were measured on a CPU node equipped with an AMD EPYC 7763 (64 cores) and 256 GB of DDR4 RAM; S-Crescendo inference times were recorded on a workstation with an NVIDIA RTX 4090 GPU. Table reports the average simulation and inference times. Across all orders, S- 3-Crescendo achieves more than two orders of magnitude speedup while preserving high accuracy.
DCM baseline. Dynamic Circuit Macromodeling (DCM) is physics-inspired model-order reduction technique that constructs dynamic macromodels to balance accuracy and efficiency. On the same 10-order RC test file, DCM completes in 0.6 with fit accuracy of R 0.9983."
NGSPICE baseline. NGSPICE is a widely used open-source SPICE simulator adopted across academia and industry. Although typically slower than commercial tools such HSPICE, its accessibility and device-level accuracy make a credible practical reference. On our 10-order RC test file, NGSPICE completes in 1.08 and-because it shares core numerical algorithms with HSPICE delivers equivalently high fidelity (we did not compute a separate R² against HSPICE).
Note. Order- 10 is the slowest inference case for our model; benchmarking against this conservative worst case still yields the above margins over DCM, NGSPICE, and HSPICE, underscoring the strength of our approach.
Table Runtime comparison between HSPICE and S- S-Crrscendo across transfer function orders (10ns, 1000 steps).
Order
1
2
3
4
5
6
7
8
10
HSPICE (S)
0.26
0.21
0.23
0.22
0.27
0.26
0.28
0.23
0.23
0.26
S- -rrsscendo (s)
0.014
0.019
0,022
0.018
0.023
0.028
0.031
0.034
0.039
0.042
Speedup (X)
18.6
11.1
10.5
12.2
11.7
9.3
9.0
6.8
5.9
6.2
6 Further Discussion
6.1 Degradation of Accuracy at Higher Orders
While S-Crescendo performs well on low to mid-order transfer functions, its R2 degrades as order m increases. This is due to recursive еπтоr accumulation: each correction module Eu adjusts not only current residuals but also propagates previous errors. If the single-pole state space has size N, then the full m-pole space grows as Nm whereas each error model sees only N samples during training. Thus, the fraction of covered states is shown Equation (12):
N m N'
(12)
which shrinks rapidly with m, leading to sparse supervision and compounded inaccuracies.
6.2 Reducing Data Dependency via Blocked Recursion
To alleviate error accumulation and data explosion, propose blocked recursion strategy. Instead of training a separate module per order, we group adjacent orders into blocks-for example, shared module for orders 2-4, another for 5-6, etc. Each block is supervised O(N B) states (for block size B), yet extrapolates across B orders. This reduces the number of recursive calls and lowers training demands, improving both runtime and scalability.
All DCM runtimes measured machine with AMD EPYC 7763 (64 cores), 1 TB RAM, and 30 GB swap.
NGSPICE runtime was measured on machine with Intel Core 17-1465UHX CPU and 32 GB RAM. S. S-Crrscendo inference uses the same RTX 4090 workstation described in this paper.

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6.3 Scaling of Inference Latency with Order
S-Crescendo's 's inference latency scales roughly linearly with order m, since each pole adds a forward pass. In contrast, tools like HSPICE collapse high-order dynamics via model reduction, maintaining near-constant runtime. bridge this gap, we consider (i) collapsing low-impact poles into aggregate corrections, (ii) integrating model-order reduction into the learned pipeline- strategies aim to extend our efficiency gains to large-scale systems.
6.4 Modeling Limitation: Repeated Poles in Transfer Functions
The model currently does not handle repeated poles, which introduce higher-order temporal terms like t To address this, future models can extend input features include pole multiplicity, enabling learning from triplets (pi, Ai, m:). This would broaden the model's applicability to more complex, higher-order dynamics.
Outlook: Toward General Nonlinear-LLinear Hybrid Systems
Beyond RC modeling, the proposed framework extends naturally to hybrid systems with "nonlinear front-end linear dynamic core" architecture, common across engineering domains. Examples include switching converters in power electronics, where nonlinear control drives linear filters; analog front-ends, where transistor drivers interface with RC loads; and neural membrane models coupling nonlinear ion channels capacitive elements. By modeling linear dynamics via Laplace-domain priors and learning nonlinear corrections, the framework enables efficient, interpretable emulation. Future directions include integrating operator learning domain-specific constraints broaden applicability.
7 Acknowledgments and Disclosure of Funding
This work was supported by the National Natural Science Foundation of China (NSFC) under Grant 62374189 and Guangdong Innovation and Entrepreneurship Team Project under Grant Innovation Strategy (2021ZT09X070).
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Appendix Contents
A. Related Works
A.1 Background
A.2 Traditional Methods for Signal Line RC Response Modeling
A.3 Current Applications of Neural Networks in Modeling Nonlinear
Systems
B. Dataset Preparation Details
B. Reference Simulator for fround-Trutte Generation
B.2 Dataset Computing Environment
B.3 Dataset Preparation
* B.3.1 Netlist Generation
* B B.3.1.1 Circuit Structure
* B.3.2 Automated HSPICE Simulation and Parser
* B.3.3 Function Extraction
1. Netlist Parsing
State StaeeSpace Model Construction
3. Transfer-Function Computation
Result Serialization
B.4 Data Alignment
C. Log-Centereed Time Warping & Resampling
C.I Motivation
C.2 Center Detection
C.3 Log-Centred Warping
C.3.1 Slope Compression
C.4 Uniform Resampling in the Warped Domain
C.5 Effect of Uniform Resampling on Regression Accuracy
C.6 Thcoretical Impact on Learning
C.7 Proof of Lipschitz Constant Reduction
Composition Lemma
* 2. Derivative of the Inverse Warp
* 3. Parameter Choice for Contraction
* Impact on Generalisation
C.8 Implementation Details
D. Full Test Case
DD.1 Training Data Preparation
D.2 First-Order Model Training and Error-Model Generation
D.3 Performance Evaluation
E. 200-Order Extrapolation Results
E. Experimental Setup
E.2 Aggregate Metrics and Case Studies
- E.3 Discussion and Limitations
15

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A Related Works
A.1 Background
Modeling and simulation of high-order nonlinear systems remain central challenges in modern VLSI backend design, particularly in analog/mixed-sigman verification and large-scale circuit time-domain modeling. Mainstream industrial tools such HSPICE predominantly rely on Newton-Raphson iterative solvers coupled with numerical integration schemes like Gear or Trapezoidal methods. While these solvers offer high physical accuracy, their simulation time and resource consumption scale poorly with system complexity-especian possing significant computational bottlenecks for practical deployment.
in the presence of strong nonlinearities or high-order
In recent years, neural networks have demonstrated remarkable success across a wide range of scientific computing applications, including areas within electronic design automation (EDA) such as device modeling [37], analog circuit fault detection [38], and high-frequency electromagnetic simulation [39]. However, for the specific task of modeling signal line RC responses in high- High-order nonlinear systems, effective deep learning-basse based approaches remain largely unexplored. To the best of knowledge, existing work has leveraged deep neural networks -articcllay Transformer-based architectures-for direct time-domain modeling of such systems.
Given this gap, we provide comprehensive review of related work from two perspectives: (I) traditional modeling approaches developed in the EDA community for approximating RC signal line responses, including current source models, voltage response models, and direct waveform fitting methods; and (2) recent advances in neural network-based methods for modeling general nonlinear dynamical systems, which-while related in scope-targe different modeling granularities and lack direct applicability to high-order S-domain waveform prediction.
A.2 Traditional Methods for Signal Line RC Response Modeling
Existing methods for modeling signal line RC response can be broadly classified into three categories:
The first category is the Current Source Model (CSM). Criox and Wong proposed a gate cell current source model called Blade [20]. which consists of voltage-controlled current source, internal capacitance, and one-step time-shift operation. Kellor further enhanced model accuracy by introducing the KTV model [40], which considers Miller capacitance. Subsequently. Li and Acar [41] and Fatemi et al. [42] introduced input and output parasitic capacitances, modeling the output current source as function of input/output voltages, gradually incorporating nonlinear characteristics into CSM models. However, since CSM-based methods can only match fixed effective capacitances (up to two) throughout the process, the simulation accuracy of current/voltage waveforms is inherently limited
44445]]]444]4] In recent years, widely adopted industry methods such Composite Current Source (CCS) [21] and Extended Current Source Model (ECSM) [48] have established driver and receiver models for each cell to handle scenarios with nonlinear input and crosstalk. Nevertheless, CSM-based approaches still face significant challenges in matching high-order RC load characteristics, limiting their accuracy in current response prediction.
The second category is the Voltage Response Model (VRM), such as the Non-Linear Delay Model (NLDM).. Iterative methods [22] [25] [49] [50]. although capable of achieving high precision, often require substantial CPU time for convergence. Non-iterative methods [51] [52], on the other hand, rely on closed-form expressions that offer faster computation but can result in output waveform matching errors of up to 15% [26]. Furthermore, technology nodes shrink and RC loads become more complex, two-parameter fitting methods struggle to accurately capture the response curve of RC networks, limiting their applicability in high-precision simulations [53].
The third category consists of Direct Waveform Prediction Methods, such as double exponential functions [23], Weibull functions [54], and gamma functions [55], which directly fit the current or voltage response. Recently, a macromodeling method [56] was proposed that uses SPICE to extract parameters for modeling, which improves accuracy to some extent. However, these direct fitting methods are unable to predict initial overshoootunndershont effects, which become more pronounced when the input slope is large.
16

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A.3 Current Applications of Neural Networks in Modeling Nonlinear Systems
In recent years, neural networks have been widely applied to modeling and prediction tasks of nonlinear dynamical systems. Neural ODE methods [12] introduce continuous- differential equations to model system evolution, which suitable for trajectory prediction problems. Physics- Informed Neural Networks (PINNs) [11] incorporate partial differential equation constraints during training to enhance physical consistency of the model. Additionally, Reservoir Computing [57] has demonstrated excellent performance in modeling short-term behavior of chaotic systems, while Koopman operator learning [58, 59] attempts linearize nonlinear systems by mapping them into a linear space to simplify modeling. Graph Neural Networks (GNNs) [60] process complex local interactions by constructing graphs among components, and neural operator methods, such as the Fourier Neural Operator (FNO) [14] and DeepONet [13]. focus on learning mappings of high-dimensional functions. In the domain of time series forecasting, Transformer-bawee models and their variants (c.g., Informer [61]) have demonstrated strong capability in modeling long-term dependencies. Although these methods have achieved substantial progress in various scientific computing scenarios, existing works mainly focus on wwwdimensional systems, continuous trajectory modeling, or frequency-dome
PDE solving, with limited exploration in modeling high-order time-domain responses of complex circuit RC networks.
The proposed S-Crescendo framework in this work possesses three key characteristics designed to effectively model time-domain behaviors in high-order nonlinear systems. First, it explicitly separates system modes via partial fraction decomposition of the S-domain transfer function. Second, it employs distributed Transformer architecture combined with pole-residue embedding strategies, reducing the response prediction complexity from cubic O (n²)) linear (nn). Third, it introduces an attention-baa based correction operator that adaptively captures nonlinear coupled responses while modeling dominant modes. This framework combines physical interpretability with computational efficiency and demonstrates high-fides fidelity waveform fitting consistent with HSPICE simulations on validation datasets, thus filling the gap of deep learning-based modeling for high-order circuit systems.
B Dataset Preparation Details
B.1 Reference Simulator for Ground-Truth Generation
All training and evaluation datasets in this work are generated using Synopsys PrimeSim HSPICE@ U-2023.60 -3-2-.... HSPICE is widely acknowledged as the de facto industry reference for analog and mixed-signal circuit simulation, owing to its consistently strong agreement with post-silicon measurements -tyiiclll achieving within 1%-5% error across a broad range of process nodes (from 0. μm to 3 nm). It provides foundry-certiffed transistor and passive device models, enabling high physical fidelity and process portability. Moreover, its Precision Parallel simulation engine supports near-linear scalability on multi-core systems such as the AMD EPYC 7763 (64-core), facilitating efficient, high-accuracy waveform generation even for large-scale nonlinear networks. These capabilities establish HSPICE as trusted ground-truth generator for validating data-driven modeling frameworks.
B.2
Dataset Computing Environment
The experiments were conducted on a computing platform running CentOS Linux 7, an operating system known for its long-term stability and high compatibility with the source code of Red Hat Enterprise Linux (RHEL). This ensures the reliability and consistency of the system environment, facilitating reproducibility of the experimental results.
The hardware platform equipped with an AMD EPYC 7763 processor, featuring 64 physical cores with base clock frequency of 2.45 GHz and dynamic boost up to 3.5 GHz. It includes a 256 MB L3 cache and an 8-channel DDR4-3200 MT/s memory architecture, providing substantial parallel computing capability and high memory bandwidth. This configuration offers ample computational resources and efficient data handling, guaranteeing smooth execution of large-scale simulation workloads while minimizing potential performance bottlenecks.
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B.3 Dataset Preparation
The dataset was constructed through a multi-stage pipeline. First, a Python script automatically generated SPICE netlists (".sp" files) representing RC networks of varying orders. These netlists were then processed along two parallel paths. the first path. Python module parsed each netlist to compute the corresponding analytical transfer function. In the second path, all netlists were batch-simulated using PrimeSim HSPICE U-2023.03-SP2-22 to obtain time-domain simulation results. Upon completion of the simulations, another Python script extracted the relevant outputs from the Finally, the analytical transfer functions and the corresponding simulated results were aligned and merged into unified dataset, which served as the basis for subsequent model training and evaluation (see Figure 5).
HSPICE raw data files.
HSPICE Simulation
Simulation Parsar
Data Alignment
Netlist
Function
Extraction
Figure 5: Overview the dataset preparation pipeline.
B.3.1 Netlist Generation
Python script automatically generates SPICE netlists for cascaded RC networks of arbitrary order. For first-order RC circuits, we adopt an exhaustive grid-based method by varying R₁ and Ci across predefined intervals construct a comprehensive dataset that covers wide range of transfer functions. The script iterates over discrete values and generates files named testsuite, <n> mp. each invoking transient simulation (. TRAN 10ps 20ns) and including standard PDK models to ensure consistency.
For higher-order RC networks (N 1), due the exponential growth in parameter space, exhaustive enumeration becomes computationally infeasible. Instead, we construct a continuous state space defined by all possible tuples {R,,, C22 1.N and sample from this space using randomized or quasi- random strategies. Each sample corresponds to a specific configuration of RC parameters, which is then used to generate netlist representing an N Nthhorder cascaded RC system. This hybrid strategy ensures both completeness in low-order cases and scalability in high-order scenarios, providing diverse and representative dataset for model training and evaluation.
The dataset generation scripts and usage examples are available at our code in supplemen- tal mal_material.zip
Circuit structure We construct a canonical nonlinear-lined cascade by driving a linear RC network with nonlinear inverter. The nonlinear frontend is implemented standard-cell inverter in SPICE:
X1 I1 OUT vdd 0 CLKINV3_ 12TR50
Here, OUT is the inverter's output node, which serves the input to the linear RC load. In the simplest first-order configuration, the RC network consists a single resistor ccppccioo pair:
R1 OUT
NODE1
r1
C1 NODE1
c1
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To generalize this to an N-stage cascade, we chain N identical first-order sections. Each section is defined in the netlist as:
R1 OUT
NODE1
r1
C1 NODE1
0
c1
R<n> NODE<n-1> NODE<n> r<n>
C<n> NODE<n>
<<n>
where the nodes are labeled sequentially as NODEO=OUT, NODE1..... NODEN, and NODEn is grounded.
Although a single stage has the well-known transfer function
Ha(s) 1 CR.,C
the overall transfer function of the cascaded network cannot be written the simple product II (s). Inter-stage coupling-where each stage's output drives the next-requires solving the full circuit equations or deriving state-space model to obtain H(s) correctly.
Once the high-order transfer function H (s) is obtained, we apply partial fraction expansion (PFE) to decompose into modal contributions:
Tk
H(s)
8 Pk:
where and denote the poles and residues, respectively. This modal form explicitly exposes the dynamic modes of the network and forms the theoretical foundation for our S-domain-aware Transformer model in time-domain prediction of high-order nonlinear circuits.
B.3.2 Automated HSPICE Simulation and Simulation Parser
The HSPICE simulation workflow was fully automated via Python script that sequentially runs all SPICE netlists. Each simulation outputs transient voltage responses at the designated output node, saved in lis files containing detailed time-domain voltage and current waveforms.
To facilitate efficient data extraction and downstream processing, a dedicated Python parser was imple- mented to systematically extract voltage response sequences from these .lis files, A representative snippet of the extracted data shown below:
time
voltage
0. 0.000000e+00 1.8579u
1.000000e-11 7.16m
2. 2.000000e-11 1. 722
3.000000e-11 2. 54157e-02
4. 4.000000e-11 3. 262519--0
5.000000e-11 3. 3.79554e-02
6. 6.000000e-11 4.32588e-02
These extracted voltage traces serve as ground truth data for validating the analytical transfer functions and for subsequent model training and evaluation.
B.3.3 Function Extraction
To characterize the linear dynamics of cascaded RC networks, implemented fully automated Python pipeline comprising four stages: netlist parsing, state-space construction, cansfer-function computation, and result serialization.
Netlist Parsing Using regular expressions, the parser reads each SPICE file extract resistor R; and capacitor C; values. For third-order network, the script locates lines beginning with R1, R2, and applies unit-aware conversion (e.g., 15 u=100-6, Missing or malformed entries trigger exception to ensure data integrity.
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2. State-Space Model Construction Given the extracted {R,, C,). admittance matrix G E R3x3 and capacitance matrix ding are assembled for the 3-stage RC ladder. The continuous-timg state- space matrices are then computed as
A - diag G, B == -=Ciial O, OJ = [0, 0, 1].
3. Transfer-Function Computation The transfer function (s) is obtained from (A, B, C) via SciPy's ss2tf routine, yielding numerator and denominator polynomials. We apply partial-fraction expansion (residue) extract poles p; and residues rt. After filtering negligible imaginary parts, poles are sorted by magnitude, and normalized coefficients are computed as
A
A
A
Σ3
4. Result Serialization For each netlist, the tuple (A1, P1, A/2, P2. 4/2, Ps) is written as single row a CSV file. This standardized format enables downstream training pipelines to ingest model parameters directly.
B.4 Data Alignment
To prepare the dataset for sequence modeling tasks, concatenate the static cireuit features with the time-dependent voltage response while preserving the temporal dimension explicitly.
Let E 2 denote the RC feature vector extracted from an r-th order RC network (e.g., normalized residues and poles). For a third-order network, we have:
(A), Pl, A/2, P2: Aly, p3) R²
Let the voltage response sequence over I time steps be represented as a set of timestamped scalar pairs:
{(1,, 31), (12, U2), (tr, br]), where E R 4 E R.
To incorporate both dynamic and static information, we replicate the static vector x at each time step and form the augmented matrix:
t1
V1
t2
y/
x
ztt)
=
(Tx(2+)))
lT
yT
This results in a fully timestamped sequence z(l), where each row consists of:
the current simulation time ft
the corresponding voltage response ye.
and the circuit's 's physical parameters
Such format allows temporal models to condition predictions only on voltage dynamics but also on circuit- speuiffc properties.
All datasets automatically constructed using Python script that fuses the time vector, voltage response, and RC features.
C Log-Centered Time Warping Uniform Resampling
C.1 Motivation
step response n(t) typically exhibits an extremely steep leading edge followed by long, almost flat settling tail. When the raw waveform is sampled on a linear time axis, the number of informative points located on the rising edge is orders of magnitude smaller than the points on the tail. Conse- quently, regression models trained with uniform loss (e.g. MSE) tend to under-fit fit the neighbourhood of the edge and over-fit the low-slope region, yielding poor predictions for signals whose rise times differ markedly inside the same simulation window.
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C.2
Center detection
Let te be the time at which the normalised voltage first crosses 0.5:
e(te) 1/2 VOD.
te arg min u(t) |ooo
The algorithm finds te with a two-stage linear B-spline interpolation around the crossing (see find_t() in the accompanying code base).
C.3 Log-centred warping
For every sample t we define a log-centred time coordinate
T sgn(( (o) [In([ lel/To +
Ine),
(13)
with scale factor To= 10 10-10 (empirically chosen) and numerical guard 0.1. Equation (13) is odd around le and strictly monotone; consequently the mapping ( is invertible and preserves temporal order.
Slope compression.
The first derivative
dr
dt t tel + ETo
large when tel is small and diminishes as It Lel + xc. Hence points on the steep edge (|t tel < To) stretched in 7-space while the quasi-flat flat tail is compressed. In effect, the dynamic range of local slopes
dt
((t - tel sToo
dr
equalised, yielding better-conditioned learning target.
80 raw
RC 80 after
1.0
1.0
0.6
0.6
Vottge
Veltage
0.2
0.2
Log
Figure 6: Raw waveform vs. log-centred waveform.
C.4
Uniform resampling in the warped domain
After warping, we resample the trace at N equally spaced r-locations
Tmax Tmin
Tmin + KAT, = O N-1,
N
Linear interpolation in r-space is equivalent to non-uniform interpolation in the original time domain, so the final dataset allocates identical representational capacity to equal increments of r, i.e. equal log-scaled time gaps with respect to the edge.
21

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Figure 7 shows: (a) The original voltage response v(t) for the first RC channel plotted against real time t. (b) The same trace after applying the center-log warp + т and uniformly resampling in 7-space. Black circles indicate the new sample locations, illustrating the denser coverage near the rising edge and sparser coverage on the tail.
RC_ 80 raw
RC, 80 after log
1.0
1.0
Log Centered Resampled points
0.6
0.6
Vollage
Voltage
0.2
0.2
0.50
1.00
Log-Tannsorrmdd
Figure 7: Scatter plot in 7-space: before vs. after resampling.
Comparison of the original step response and the uniform resampling in log-time domain. The blue markers show the raw voltage trace v(() for the RC network sample (from Figure 8), plotted against the log-centered time coordinate The orange markers overlay the uniformly spaced samples in T-space, demonstrating that the resampling allocates more points near the steep rising edge while compressing the long tail.
RC PC_cated_110.co c40.csv
Uniform
0.9
Voltage
0.4
0.0
-4
-2
Log-Time
Figure 8: Scatter plot r-space: resampling vs. after uniformed resampling.
C,5 Effect of Uniform Resampling on Regression Accuracy
We conducted an experiment to quantify the impact our log-centered warping plus uniform resampling on step-response modelling accuracy. Under identical training conditions: 60 epochs, batch size 1000, using 80 training files (80 1000 samples) drawn from the full set of 100 RC cases. The model was evaluated on the held-out test set comprising cases RC_81 through RC_100 We report the coefficient of determination R² for predictions made with ("Resampled") and without ("Raw") the uniform resampling in warped time.
22

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Table It' comparison: raw vs. log-centered 7-uniform resampling.
Case
R² Raw
R² Resampled
RC 81
0.9571
0,9998
RC 82
0,9591
0,9998
RC 83
0,9609
0,9998
RC 84
0,9629
0,9998
RC 85
0.9647
0,9998
RC 86
0.9662
0,9998
RC_87
0.9675
0,9998
RC_88
0.9690
0,9998
RC
0.9702
0,9998
RC 90
0.9714
0,9998
RC 91
0.9725
0,9998
RC 92
0.9736
0,9998
RC RC_93
0,9745
0,9998
0.9753
0,9998
0.9761
0,9998
RC 96
0.9768
0,9998
RC 97
0.9774
0,9998
RC_98
0.9780
0.9998
RC 99
0.9785
0,9999
RC 100
0.9816
0,9999
Mean
0.9715
0.9998
Qualitatively, Table shows that uniform resampling in r-space consistently boosts R² from the high- 0.95 range up to nearly perfect 0.999. On average, the preprocessing yields an absolute improvement of over 0.028 in R2, demonstrating that our log-centered warp and uniform sampling dramatically enhances the model's ability to fit diverse step-response curves under identical training regimes.
As shown Figure we compare the predictive accuracy of our network on the same RC trace under two preprocessing regimes. In subfigure 9a, the model is trained and evaluated directly the raw time-domain samples, achieving an lt² of 0.9571. While the overall step response shape is captured, the prediction exhibits noticeable lag the steep rising edge and slight deviation in the mid-tail region. In contrast, subfigure 9b illustrates the result after applying the center-log warping followed by uniform resampling in the warped 7-domain. Here, the fit improves dramatically to R² of 0.9998, with the predicted curve (orange) virtually indistinguishable from the ground truth (blue) across both the fast edge and the extended settling tail. This comparison clearly demonstrates that our log-centering and uniform resampling pipeline substantially enhances the model's I's ability to learn and generalize step-response dynamics under identical training conditions.
SC cabed 11 55-51511
SC cabed EL 51-0-3000
1.
0.3
0.7
03
1ime
10-4
(a) Original time-domain fit (R"=0.9571).
(b) Log time fit after preprocessing (R-=0.9998).
Figure 9: Model predictions vs. ground truth for two RC traces: (a) raw time-domain input, (b) preprocessed (center-log warp + uniform resampling) input.
23

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C.6 Theoretical impact on learning
Variance reduction. Let oew denote the variance of the target derivative on the raw grid and wamp the variance of du on the warped grid. From the slope-compression term above follows that warp I raw. reducing heteroscedasticity seen by the leamnrr
dr
Effective sample size (ESS). The warping acts as an importance saaoplinc-ssmppiigg scheme with weight (()) = tel + eTo ESS increases because highly informative edge samples receive larger weights after transformation.
Improved Lipschitz constant. Denote the network fo with Lipschitz constant L. The composite function fo oo (where g the inverse warp) retains the same L but is now evaluated domain where the target variation is smaller, tightening generalisation bounds.
C.7 Proof of Lipschitz Constant Reduction
We now show that, under mild parameter choices, the pre-/post-warp composite F(r) fo (7))
has an overall Lipschitz constant I' L.
Composition lemma. f is Lipsschht and h: Z - X is Lipsccit, then [I o h(z1) - fo h(2))l & (1(()) - h(22)l s Ly Ln || 22ll-
so f h is I [62].
Derivative of the inverse warp. Recall the forward warp
== gt) = sgn(t te) Inn [t-tel
+ E) In
To
Differentiating,
dr
dt
It tel To.
dt
1t tel fol+&8 &
dr
Since a one-dimensional Theorem [63], follows that
C function with sup [M(x)] s M is M-Lipschitz by the Mean Value
L Lg
sup TE(TmiinsTman
dg
sup
(t- tel TT).
tE f
Parameter choice for contraction. parameters that
Let At = maxx(lmax tel. Itmin - tel)- we choose
A + & To <1,
then
Lg-1 <1
='= Ly-1 < Life
Hence F(r) is strictly more contractive than fo on the original I-domain.
4. Impact generalisation. Standard Rademacher--omplexxi generalisation bounds scale linearly with the Lipschitz constant of the hypothesis class and inversely with VN [64, 65]. By reducing the effective Lipschitz constant from to L' < L, we tighten the bound
Egen
C
V
thereby improving expected test performance.
C.8 Implementation details
1. Voltage normalisation. All voltages are sealed by VoD so that u E [0,
2. Time origin. Warping performed after shifting the origin t.. This removes sample-to- sample phase variation.
3. Parameter choices. To controls the width of the expanded region; our experiments
(10-103) s) adequately covers modern technology nodes down to 3 ps edges.

---

D
Full Test Case
D.1
Training Data Preparation
Netlist Generation
Prepare pool of base RC parameter pairs
Run:
python prepare/n.py --order n --count
output_dir sp_files_n
HSPICE Simulation
Simulations are executed via sim. sh under Linux.
Run:
hspice testsuite_<n> <p> -o lis/result_cnp <n>
Simulation outputs are stored in the lis/ directory.
Simulation Parser
Place your HSPICE lis files into the //result/ directory.
Run the parsing script extract time- time-voltage waveforms and convert units:
python parse_spice, py \ --input_dir ./result --output_dir ./result/csv/
Parsed CSV files will be saved in //tssut/css// for further analysis.
Function Extraction
Place your SPICE netlist files (.sp) into the directory sp_files_n/.
Run the extraction script to parse resistor and capacitor values, compute state-space parame- ters, poles, and residues:
python ietract_functios py \
--input_dir sp_ ap_files_n --output_dir result_n
The extracted normalized residues A& and poles p; for each netlist saved CSV files in result_n/.
Data Alignment
Place your RC parameter CSVs in RC/ and parsed SPICE result CSVs in lis/result/.
Run the script to merge each pair into result/ as follows:
python data_ al awiggnentt.pp py
This script reads the first row from each RC CSV, duplicates it to match the length of the corresponding SPICE CSV, concatenates them column-wist and saves the combined CSV.
Processed files are named BC_ cC__cate_ii i.cvv
csv for 1,...,
25

---

D.2 First-Order Model Training and Error-Model Generation
The following workflow describes how to train the first-order predictor and automatically generate the residual ("error") datasets for higher-order correction models. All commands assume you are in the repository root (see code in papplementl__matrialllll
1. Log-centred warp & resampling for Model data
Place your pre-factorised first-order training CSVs into
Run:
python testxiao/testxiaa py
--input_die dir testxiaoobasiccmooee \
--output_din _dir testxiac/results
--warp center-log
Intermediate warped traces appear in testxiao/resultts/
Next, uniformly resample in 7-space:
python testxiaoouniifor.ppe \
--input_dir testxiao/resultt
-outpptt_irr data/model1
The resampled CSVs for Model I are now in data/model1//
2. Train the first-order model
Launch training:
python train_ maain_modell.pp \
--data_dike data/model1
--save_path
Upon completion, the static first-order model is saved models/static_cond__ne sacpatiiiccoo_mmmeelleee_ mth.
3. Generate error datasets
Execute the pipeline script to compute residuals:
python pipeline_ aeteeerrii_eeeee error satror_mdels...mm py
--model_path
http:/sttticccooooooooo coondmmddel.pp pth --input_dir data/modell \
--tmp_dir _tmp_ds
This creates error-of-order-2 and error-of-order-3 traces in tsmp_s// Copy the intermediate error files into the next data folder:
cp ttp_ttpppaaateet aas/eerrrr... * csv data/error/ error/
4. Train the error models
Finally, train the error prediction networks:
python combine combine_and_tte htt__taiineerrr_modeee.
--data_das data/error \ --svve_pret --vve_prefix models/model_ems
This produces
pth and models/mooee__eror..33 which predict the 2nd- and 3rd-order residuals respectively.
All script invocations include detailed usage notes in their headers. For full examples and parameter options, consult the source files in the testxiao/ and root directories our code supplemen- tal material.zip
26

---

D.3 Performance Evaluation
Once the static and enror models have been trained (models/static, ceteaatiicoond_eeeeeeeeeee sandd_mode.pg pth and models/model, amooel_error__a error_n. pth), we run the recursive inference script to generate test predictions and metrics. You may adjust inference parameters (e.g. batch size, lookback window) via command-line flags.
python test_ aest_rrcursive__ifeeeneee \
--data_diz data/order1/test
static_model models/staaiiccondd_mmm moddelel...m
--err_mode --err_model models//mdel__errr..n..o
--out_dir results
The script outputs:
Predicted vs. true response CSVs in results/
Summary (R2) printed to console and saved under results/metrics.
Plots of step-response comparisons in results/plots/
E 200-Order Extrapolation for Dynamic Signoff
To address the dynamic signoff mode, we conducted high-order experiments to 200 poles. We trained residual-correction modules systems with no more than 15 poles and evaluated on 200-pole network. The model achieved a coefficient of determination of H²² = 0.9838, indicating that our approach scales orders required for full dynamic signoff with a still manageable training cost (approximately 5-6 hours on single RTX 4090).
E.1 Experimental Setup
We first generated training data for systems of order < 15 using standard HSPICE simulations. Residual-correction modules contained approximately 100-160K parameters and were trained with mixed precision on a single NVIDIA RTX 4090 GPU. Hyperparameters followed the settings used in lower-order experiments (optimizer, learning rate schedule, batch size), with early stopping based on validation R The 200-pole evaluation network was held out from training and tuned to match the transient signoff configuration (10 ns horizon, 1,000 samples).
E.2
Case Studies
report aggregate metrics and qualitative comparisons against HSPICE on the 200-pole network. The model attains n2 = 0.9838 with low mean absolute error and well-behaved residuals across the transient window. Visual overlays show close alignment both fast and slow time scales, with the largest errors concentrated near rapid slope changes.
27

---

RC RC_cated R =0=0.9838
1.0
0.9
0.8
veltage
0.7
0.6
0.5
0.4
-4
-2
2
6
log time
Figure 10: 200 order case study
E.3
Discussion
These results demonstrate that the residual-correction design generalizes beyond the training order range, providing stable extrapolation to 200 poles. While inference remains fast and training cost scales primarily with data volume and sequence length; the reported 5-6 hours is one-time training cost and remains practical for signoff contexts. Remaining gaps are localized to high-curvature regions, suggesting that targeted augmentation (e.g., emphasizing rapid transients) and modest architecture scaling could further close the accuracy margin without compromising the observed efficiency benefits.
28

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Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
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35


Reviews:
[{"review_id": "EXlDHeI3UA", "reviewer": "Reviewer_F5ru", "summary": "This paper presents S-Crescendo, a novel, physics-informed deep learning framework for simulating high-order nonlinear systems, with a specific application to RC interconnect networks in VLSI design. The core idea is to leverage the formal structure of linear systems theory by decomposing a high-order transfer function into its constituent first-order pole-residue pairs in the S-domain.", "questions": "Could the authors comment on why other relevant baselines (e.g., a \"vanilla\" Transformer baseline that takes all poles/residues as a flat input, a NeuralODE, or an FNO) were not included?\n\nHave you performed any analysis to characterize how the error propagates as a function of model order n?", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- Instead of treating the system as a complete black box, the authors use established principles (partial fraction expansion) to structure the learning problem. This physics-informed approach is a promising direction for building more robust, generalizable, and interpretable models for physical systems. The application to VLSI simulation is well-motivated and addresses a real-world computational bottleneck.\n\n- The demonstrated speedup over HSPICE is substantial and practically relevant. By bypassing iterative matrix solves (e.g., Newton-Raphson for the Jacobian), the method offers a scalable alternative for rapid design space exploration and verification, where thousands of simulations are often required. The complexity analysis, showing a shift from O(n^3) to O(n), is a key highlight.\n\n- Strong Empirical Results: The model achieves impressive accuracy on the presented tasks. The near-perfect fit on single-pole systems (Fig 3a) validates the baseline model, and the effectiveness of the recursive correction is clearly shown (Fig 3b). The generalization experiments (Table 1) provide evidence that the model can extrapolate beyond its training distribution, which is a non-trivial achievement.\n\nWeaknesses:\n- The core innovation—the recursive training of residual modules—is also my primary concern. The authors rightly acknowledge in Section 6.1 that this approach is susceptible to error accumulation. As the order n increases, each module e_n is tasked with correcting the output of a chain of n-1 previous predictors. Any inaccuracies in the early stages will inevitably compound. While the results up to order 10 are strong, it is not clear this approach would remain stable for significantly higher-order systems (e.g., n=50 or n=100), which are common in real-world interconnect models. The proposed mitigation of \"blocked recursion\" (Section 6.2) is an interesting idea but is presented as future work and is not validated.\n\n- The only baseline is HSPICE, which serves as a \"golden\" reference and a benchmark for speed. However, there is a large body of work on data-driven system identification, including other neural network approaches (e.g., NeuralODEs, FNOs, PINNs, as cited by the authors). A comparison against at least one state-of-the-art ML-based method would help situate the performance of S-Crescendo within the broader landscape and better justify the architectural choices made.", "was_revised": true, "final_justification": "The author gives detailed answer on my questions which further strength the paper.", "rebuttal": "## Response to Reviewer: Missing Baselines, Error Accumulation, and Scalability\n\n### 1. Missing Baselines (e.g., NeuralODE, FNO, Flattened Transformers)\n\nA very insightful question. We would like to clarify that our work is conceptually distinct from many existing machine learning approaches that focus on learning equation or conversion, such as NeuralODE or FNO.\n\nMost existing ML-based solvers (e.g., PINNs, NeuralODEs, FNOs) rely on prior knowledge of the exact system equations and are tightly coupled to the specific form and scale of the dynamics. As a result, their generalization across different systems is often limited and requires re-deriving or re-solving equations for each new task.\n\nIn contrast, our approach avoids solving system equations altogether. Instead of approximating the solution to a given equation, we directly learn the transfer function that characterizes system behavior in the frequency domain. This enables a more modular and reusable learning paradigm that decouples model training from equation solving, offering better scalability and generality across a wide class of systems without requiring system-specific supervision.\n\n**Response Regarding the Absence of Vanilla Transformer, NeuralODE, and FNO Baselines:**\n\nRegarding the mentioned baseline models, we believe that due to fundamental differences in modeling assumptions and input structures, they are not directly applicable to our problem setting. The reasons are as follows:\n\n* *Vanilla Transformer* with a flat input of poles/residues essentially reduces to a matrix pre-training setup, requiring huge datasets to learn order-dependent weights in the transfer function; in our preliminary experiments with several samples, this formulation underperformed and generalized poorly, and we will include the results in the revision.\n\n\n\n* *NeuralODE* assumes continuous-time autonomous dynamical systems and learns state evolution via differential equations. In contrast, our inputs consist of discrete modal features and the system behavior is generally non-autonomous, making NeuralODE difficult to directly apply in this context.\n\n* *FNO* is primarily designed for function-to-function mappings defined over continuous spatial grids. Our modal features, however, are discrete and sparse parameter sets lacking such spatial continuity, rendering FNO not directly transferable.\n\nFor these reasons, we have focused on demonstrating how physically structured representations, such as modal decomposition in the S-domain, can be effectively interpreted with modern sequence models to better capture the coupling between nonlinear and linear system dynamics. Compared to treating the system as a black box with generic function approximators, this approach significantly enhances model generalization. We believe this direction offers valuable insights for modeling in EDA and plan to incorporate additional baselines in future work to provide more comprehensive comparisons.\n\n---\n\n### 2. On Error Propagation with Increasing System Order and Model Scalability\n\nWe value the reviewer’s concern regarding potential error accumulation as the system order \\( n \\) increases. To illustrate this, we provide theoretical proof and experimental evidence demonstrating that **although higher-order modes are not negligible, their cumulative error tends to be effectively suppressed due to their rapidly decaying time-domain influence**. As a result, the overall approximation error exhibits convergent behavior rather than unbounded disaster.\nT**Theoretically.** The modal decomposition of a linear system’s transfer function exhibits a frequency–time duality under the Laplace transform:\n$$\nH(s) = \\frac{r_1}{s + p_1} + \\frac{r_2}{s + p_2} + \\cdots + \\frac{r_n}{s + p_n}\n\\quad \\xleftrightarrow{\\mathcal{L}} \\quad\nF(t) = r_1 e^{-p_1 t} + r_2 e^{-p_2 t} + \\cdots + r_n e^{-p_n t}.\n$$\nHere $p_i$ (with $0 < Re(p_1) \\le \\cdots \\le Re(p_n)$) and $r_i$ denote the $i$-th pole parameter and residue; stable systems have poles at $s = -p_i$ and time constants $\\tau_i = 1/ Re(p_i)$. Lower-order poles (smaller $Re(p_i)$) govern dominant dynamics, whereas higher-order poles (larger $Re(p_i)$) correspond to fast-decaying components that primarily affect the very early transient. Consequently, as $Re(p_i)$ grows, the term $e^{-p_i t}$ decays more quickly, exponentially suppressing the time-domain impact of higher-order modes. Small modeling errors in these terms are therefore strongly attenuated in the output, creating a balancing effect that drives error convergence.\n\nThis diminishing-contribution effect is analogous to a Taylor expansion near $x \\to 0$:\n\n$$\nf(x)=f(0)+f'(0)x+\\frac{f''(0)}{2!}x^2+\\cdots+\\frac{f^{(n)}(0)}{n!}x^n+\\cdots,\\quad x\\to 0,\n$$\nwhere higher-order terms contribute progressively less. This same principle underlies **model order reduction (MOR)**: retaining only 5–10 dominant poles captures the primary system dynamics [1, 2, 3], while the residual due to neglected higher-order modes remains bounded and self-limiting over time.\n\n**Industrial context: two complementary modes.**  \n- In **static analysis**, designers perform rapid pass/fail checks against timing constraints by retaining only the first 5–10 dominant poles. In this regime, the truncated time-domain response\n  $$\n    F(t) \\approx r_1 e^{-p_1 t}+r_2 e^{-p_2 t}+\\cdots+r_{10} e^{-p_{10} t}\n  $$\n  closely approximates the full waveform because poles are ordered by damping; higher-index modes decay faster and contribute less—akin to the diminishing higher-order terms in the Taylor analogy.\n\n- In **dynamic signoff**, high-fidelity prediction of power, voltage drop, and reliability often requires **hundreds of poles** to capture fine-grained effects.\n\n**Experimentally.** Our measurements corroborate the above intuition. To be precise, we report the **incremental error contribution** of the $i$-th mode, defined as the **dataset-averaged change** in the evaluation error when moving from $i-1$ to $i$ modes:\n\n$$\n\\Delta \\epsilon_i := E_{data}[ E_i - E_{i-1} ].\n$$\n\n\nwhere ${E}_i$ denotes the output $\\ell_2$ error of the $i$-mode model. \nThe values below are the **dataset means** of $\\Delta \\varepsilon_i$; \nthe sign indicates the direction of change (negative $=$ error decreases), \nand the magnitude reflects the size of the incremental contribution:\n\n$$\n\\text{2nd: } -0.02995,\\quad \\text{3rd: } 0.01362,\\quad \\text{4th: } -0.00609.\n$$\nAcross runs, $\\lvert \\Delta \\varepsilon_i \\rvert$ is typically 2--4 $\\times$ smaller} than $\\lvert \\Delta \\varepsilon_{i-1} \\rvert$, showing a clear trend of diminishing incremental error from successive higher-order modes: their influence is **not zero**, but is progressively reduced and effectively neutralized by time-domain decay, thereby preventing cumulative error buildup.\n\n\n**Scalability across the two modes.** Our initial submission reported results consistent with the **static analysis** regime (5–10 poles). To address the **dynamic signoff** regime, we conducted **additional experiments on a 200-pole RC network**. Training on systems up to 50 poles and evaluating generalization on an unseen 200-pole system yielded high fidelity with ($R^2=0.9831$) on the output waveform. When **extrapolating** beyond the training order (from 50 to 200 poles), we did observe signs of **cumulative error** as the modal count increases—consistent with the reviewer’s concern and reflecting a realistic limitation of our current recursive design at very high orders.\n\n**Practical takeaway.** For **static analysis**, we therefore adopt **10th-order reduced models** as a **deliberate, practical compromise**, balancing accuracy, generalization, and compute. At the same time, the 200-pole experiment demonstrates that the framework can operate in the **dynamic signoff** setting when high-order fidelity is required—consistent with MOR practices in commercial EDA tools (e.g., Cadence Spectre, Synopsys PrimeTime), where 5–10 poles are typically sufficient for timing, while higher orders may be invoked for signoff accuracy [1, 2, 3].\n\nWe will include **convergence plots** (e.g., output $\\ell_2$ error vs. system order) and the **200-pole dynamic-signoff waveforms** in the revised manuscript (supplemental materials), together with an analysis of how cumulative error scales with order under each industrial mode.\n\n------\n### References\n\n[1] Odabasioglu, Celik, Pileggi. *PRIMA: Passive reduced-order interconnect macromodeling algorithm.* Proceedings of IEEE ICCAD, 1997: 58–65.  \n[2] Antoulas, A. C. *Approximation of Large-Scale Dynamical Systems*. SIAM, 2005.  \n[3] Freund, R. W. *Model Reduction Methods Based on Krylov Subspaces*. Acta Numerica, 2003, 12: 267–319."}, {"review_id": "GBoAFmlbvE", "reviewer": "Reviewer_SmgB", "summary": "S-Crescendo models nonlinear systems by decomposing their transfer functions into poles and residues, which are used to construct a feature representation. A baseline prediction is made using a Transformer on first-order coefficients from the transfer function. Then, residual models, consisting of lightweight Transformers, iteratively refine the output by minimizing accumulated MSE based on prior predictions.", "questions": "The paper mentions the limitations of classical techniques such as CSMs, VRMs, and direct waveform prediction models (lines 52–54). Have the authors conducted a direct empirical comparison between these methods and S-Crescendo on the same benchmark tasks? \nThe corrective residual function is described as a “lightweight Transformer” (line 180), but Transformers are generally considered computationally expensive. Can the authors provide training time, number of parameters, and compute usage for the full model?  \nIn Table 2, the authors compare S-Crescendo’s runtime solely to HSPICE. Are there other simulation tools the authors could compare against to strengthen their claim?", "limitations": "see above", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Quality: The introduction and theoretical background sections provide a strong foundation for the rest of the work and the algorithm as well as its limitations are clearly explained. The motivation for using a Transformer is well justified: due to its ability to capture long range dependencies and process complex interactions (lines 56-57). The authors also cite the limitations of classical modeling techniques.\nClarity: The paper is generally well written. Figures 1 and 2 compliment the recursive training procedure section well and enhance understanding of the S-Crescendo algorithm. The rationale for design decisions, such as using a Transformer for non-analytic nonlinear responses, is well-articulated (e.g., lines 162–163). However, there are numerous word spacing and formatting issues throughout the text, which slightly detract from readability.\nSignificance: S-Crescendo successfully improves the efficiency of non-linear system modeling: a relatively unexplored direction of circuit design. Compared to HSPICE, the framework achieves faster runtime performance and maintains accuracy, which is promising. However, the evaluation lacks depth in benchmarking against a broader range of classical tools beyond HSPICE as well as classical model prediction techniques. \nOriginality: The paper presents an original approach by integrating deep learning and Transformer architectures into nonlinear circuit modeling in the S-domain. S-Crescendo  produces a runtime advantage compared to other well-known software, as shown in Table 2. However, a more explicit comparison to classical methods as well as a comparison to a diverse survey of simulation software would make the authors’ claim more compelling.", "was_revised": true, "final_justification": "This is a nice paper and authors responded to reviews by improving it. I support acceptance.", "rebuttal": "## Response to Reviewer:\n\n### 1. Expanded Baselines (Comments 1 and 3)\n\nWe accept the reviewers’ suggestion to expand our comparative study. In the revised version, we will include additional benchmarking against the A Dynamic Capacitance Matching (A state-of-art algorithm evolved from CSM) method and NGSPICE.\n\n**DCM** is a physics-inspired, model-order reduction technique used to construct dynamic macromodels that preserve both accuracy and efficiency. On the same 10-order RC test file, DCM finishes in **0.6 s** with a fit accuracy of \\(R^2 = 0.9983\\).\n\n**NGSPICE**, by contrast, is a popular open-source SPICE simulator widely used in both academia and industry. Although slower than commercial tools like HSPICE, its accessibility and device-level accuracy make it a credible and practical reference. On our 10-order RC test file, NGSPICE completes in **1.08 s** and—since it shares core algorithms with HSPICE—delivers equivalently high fidelity (we did not compute a separate \\(R^2\\) against HSPICE).\n\n**S-Crescendo (ours)** achieves substantially faster inference on the same 10-order RC test file, completing in **0.042 s**, while maintaining the accuracy characteristics reported in our experiments.\n\n\nAll runtime measurements for **DCM** were performed on a machine with CPU: AMD EPYC 7763 64‑Core Processor and memory: 1 TB RAM + 30 GB swap. **NGSPICE** is perform on a machine with CPU: Intel(R) Core(TM) i7-14650HX and memory: 32Gb. For our model, the experiment is based on the same hardware described in our paper.  \n\n\nBy incorporating comparisons to DCM and NGSPICE under identical test conditions, we aim to provide a broader and more convincing evaluation of S-Crescendo’s effectiveness across modeling paradigms.\n\n---\n\n### 2. On Transformer Complexity (Comment 2)\n\nOur term “lightweight Transformer” refers to a simplified but delicate architecture with constrained parameters and computations. Specifically, each residual correction module contains only a few attention layers with a small hidden dimension (64), resulting in a total parameter count around 100K. Moreover, these modules operate on modal features (pole-residue representations) rather than high-dimensional raw waveform data. This design significantly reduces computational overhead while effectively capturing nonlinear system dynamics.\n\nTo clarify the training cost:\n\n- The total number of parameters is approximately 100K–160K for each residual module.  \n- We use a single NVIDIA RTX 4090 GPU for all experiments.  \n- For single-pole models, the full training process takes less than 20 minutes, demonstrating both low cost and excellent generalization.  \n- For multi-pole systems (e.g., 10th-order models), the training time is around 2 hours per experiment, which remains feasible given the significant speedup it offers during inference.\n\nTherefore, while our architecture is Transformer-based, it is delicately tailored for efficiency and scalability, with training costs that are practical for both research and industrial applications.\n\nWe will add an extra section in the revised manuscript reporting training runtimes, GPU utilization, and dataset-construction overhead, with full details provided in the supplementary material to ensure transparency and reproducibility."}, {"review_id": "0XUOOvkjP1", "reviewer": "Reviewer_o5vd", "summary": "The paper proposes S-Crescendo, a physics-guided machine learning framework for fast nonlinear circuit simulation, combining S-domain modal decomposition with transformer-based residual learning. The proposed method formulates circuit dynamics as a superposition of interpretable pole-residue pairs, enabling linear complexity inference. It then introduced a nested transformer architecture to compensate for nonlinear coupling effects without sacrificing physical consistency. The method targets rapid design space exploration, particularly for parameterized nonlinear systems.", "questions": "- The paper asserts suitability for \"large-scale\" problems but only tests systems up to 10th-order. Would you provide results on at least one industrial-scale testcase?\n- What is the training cost and the cost to build training set?", "limitations": "- Training cost is unaddressed.\n- The Transformers memory footprint on large-scale instances is unaddressed.", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The method is technically sound to me.\n- The proposed hybrid approach is more tractable than PDE systems, and more advanced than pure black-box surrogates (e.g., PINNs) or linearized approximations.\n- Experiments validate the claimed efficiency gains for small-to-medium systems.\n\nWeaknesses:\n- The scalability claims lack empirical validation. While theoretical complexity analysis suggests O(n) scaling, the absence of large-scale benchmarks undermines the paper’s relevance to industrial-scale problems.\n- I am concerned on the training cost. Wouldn't the training data dependent on HSPICE either? I would not trust on the extrapolation of a model solely trained on small-scale instances.\n- It seems not suitable for strongly coupled nonlinear systems, but I am not sure.", "was_revised": true, "final_justification": "The rebuttal addressed my concerns. I do appreciate the supplemented materials including large-scale performance, training runtime, GPU usage, and dataset construction overhead.", "rebuttal": "## Response to Reviewer:\n\n### 1. “Large-scale” applicability and 10th-order evaluation scope\n\nIt’s a very incisive question. Actually in semiconductor industry, there are two prevalent modes for handling timing and waveform simulation:\n\n1. **Static Analysis** for quick pass/fail checks against timing constraints and signal integrity by using reduced models.\n2. **Dynamic Signoff** for accurate prediction of power consumption, voltage drop, and reliability variation, which often requires high-fidelity, high-order models.\n\nIn our first submission, we demonstrated our performance under the **Static Analysis** mode, where only the first 5–10 dominant poles are retained. In this regime, the frequency–time duality yields\n\n\\begin{aligned}\nH(s) &= \\frac{r_1}{s + p_1} + \\frac{r_2}{s + p_2} + \\cdots + \\frac{r_{10}}{s + p_{10}} \\\\\n&\\xleftrightarrow{\\mathcal{L}} \\quad\nF(t) = r_1 e^{-p_1 t} + r_2 e^{-p_2 t} + \\cdots + r_{10} e^{-p_{10} t}.\n\\end{aligned}\n\n\nThis truncated representation closely approximates the full waveform \\textbf{because we sort the poles} so that\n$0 < \\operatorname{Re}(p_1) \\le \\operatorname{Re}(p_2) \\le \\cdots \\le \\operatorname{Re}(p_{10})$\n(stable systems have poles at $s=-p_i$). As a result, higher-index poles have \\textbf{larger damping}\n(shorter time constants $\\tau_i = 1/\\operatorname{Re}(p_i)$), and their time-domain contributions\n$r_i e^{-p_i t}$ \\textbf{decay faster}, thus exerting progressively less influence on the overall waveform.\nThis is analogous to a Taylor expansion near $x \\to 0$, where higher-order terms contribute less.\n\n$$\nf(x) = f(0) + f'(0)\\,x + \\frac{f''(0)}{2!}x^2 + \\cdots + \\frac{f^{(n)}(0)}{n!}x^n + \\cdots,\\quad x \\to 0\n$$\n\n\nwhere the contribution of higher‑order terms decreases.\n\nTo address the **dynamic signoff** mode, we have now conducted **200‑pole experiments**. We trained on systems up to 50 poles and tested on a 200‑pole network, achieving ($R^2$ = 0.9831). This demonstrates that our method can scale to the very high orders required for full dynamic signoff, at the expense of still manageable cost (5-6 hours training).\n\nBy explicitly presenting both the 10‑pole static analysis results and the new 200‑pole dynamic signoff results, we clarify how our approach flexibly supports both industrial workflows:  \n- **Fast Static Analysis** with 10‑pole models,  \n- **Accurate Dynamic Signoff** with 200‑pole models.\n\n*(Note: due to space constraints, the full 200‑pole comparison plots will be included in the revised manuscript’s supplemental materials.)*\n\n---\n\n### 2. Training cost and data generation overhead\n\nRegarding the practicality of training and dataset preparation, we provide a detailed explanation of both components below.\n\n**Training cost.** Each residual-correction module contains approximately 100K–160K parameters. All models were trained on a single RTX 4090 GPU. For reference, single-pole systems typically train in under 20 minutes, while 10th-order systems converge within approximately 2 hours. Given the considerable acceleration our model achieves at inference time, we believe this one-time training cost is highly acceptable.\n\n**Dataset construction.** Training data is generated using standard HSPICE simulations. Each simulation typically completes in about 0.5 seconds. For our full training dataset of ~5,000 samples, the total preprocessing pipeline (including simulation, waveform extraction, and supervised data formatting) completes in under 2 hours. The process is fully scriptable and parallelizable, and does not pose a practical bottleneck.\n\nWe will supplement the revised version with a dedicated section detailing training runtime, GPU usage, and dataset construction overhead. These details will also be included in the supplemental material to ensure transparency and reproducibility.\n\n---\n\n### 3. Suitability for strongly coupled nonlinear systems\n\nIt is a very thoughtful observation regarding the limitations of our method. We agree that our current framework is not intended to cover all categories of nonlinear system. Instead, we provide a unique way of treating a major subgroup by learning the transfer function in S domain (why our title called S-Crescendo).\n\nSpecifically, our method is designed for a widely encountered class of circuits where nonlinearities are primarily localized in the **driver**, while the **load** consists of a large but linear passive network (e.g., RC ladders, transmission lines) in which transfer function can be obtained. This pattern—nonlinear excitation followed by high-order linear response—covers over 99% sub-circuit of practical VLSI design.\n\nIn these systems, the simulation challenge stems not from the nonlinear dynamics but from the *interaction between nonlinear excitation and complex linear structures*. Although the load is linear, its high order makes repeated simulation expensive. Our model is explicitly tailored to this case:  \n1. Modeling the linear load via modal decomposition in the S-domain, and  \n2. Using residual learning to correct the nonlinear input effects."}]

Author rebuttal:

Meta-review: The paper introduces S-Crescendo, a physics-informed framework for nonlinear circuit simulation that leverages pole–residue decomposition with Transformer-based residual learning. Reviewers praised its clear motivation, technical soundness, and strong runtime improvements over HSPICE, noting its potential for VLSI applications. Concerns centered on the limited evaluation scope, absence of comparisons to other ML baselines, and questions about scalability and training cost. Overall, reviewers found it a promising and practically relevant contribution, with the majority supporting acceptance.
Decision: Accept (poster)