Amplitubes: Amp Modeling & Graph Cosmology

• Amplitubes are dual-use concepts representing both virtual amplifier systems with neural, parametric, and hybrid models, and amplitude-like functions in graph-based cosmology.
• In audio engineering, they span a design space from traditional DSP to black-box and gray-box neural approaches, balancing accuracy, interpretability, and scalability.
• In mathematical physics, amplitubes manifest in positive geometry via tubings on graph associahedra, enabling structured decompositions of cosmological correlators.

“Amplitubes” appears in current technical literature in two unrelated senses. In audio engineering and music technology, the term denotes AmpliTube-like virtual guitar amplifier systems and, more broadly, neural or hybrid emulations of analog amplifiers and effects that expose controllable tone spaces, knob-conditioned mappings, or reference-audio conditioning. In mathematical physics, “amplitubes” denotes amplitude-like functions associated to graphs that reorganize cosmological wavefunction coefficients and correlators into sums over tubing structures and positive geometries (Chen et al., 2024, Glew et al., 24 Feb 2025, Glew, 9 Jul 2025).

1. Virtual amplifier systems and the meaning of “AmpliTube-like”

In the audio literature, virtual amp systems are software or embedded emulations of guitar amplification chains that aim to reproduce nonlinear distortion, dynamic response, frequency shaping, cabinet coloration, and user-facing controls such as Gain, Bass, Mid, Treble, Presence, and Master. Traditional DSP or circuit-modeling approaches emulate each amp/cab/mic configuration parametrically, often requiring device-specific models and careful tuning; accuracy is high and controls are interpretable. By contrast, non-parametric neural captures such as NAM train one model per fixed knob configuration, giving excellent fidelity but no continuous knob control and high capture cost when many settings are required (Chen et al., 2024, Grötschla et al., 30 Sep 2025).

A recurring problem formulation is therefore whether an “AmpliTube-like” system should be built as a collection of bespoke device models, as a single parametric model conditioned on knob values, or as a one-to-many model conditioned on a learned tone representation. The literature now contains all three strategies. This suggests that “Amplitubes” is best understood not as a single implementation paradigm but as a design space spanning circuit-derived, black-box neural, gray-box differentiable DSP, and latent- or embedding-conditioned systems.

Paradigm	Representative work	Defining property
Traditional DSP/circuit modeling	(Chen et al., 2024)	Device-specific amp/cab/mic models with interpretable controls
Non-parametric neural capture	(Grötschla et al., 30 Sep 2025)	One model per fixed knob configuration
Parametric neural modeling	(Grötschla et al., 30 Sep 2025)	One model conditioned on a knob vector
One-to-many tone-conditioned modeling	(Chen et al., 2024)	Single renderer driven by a learned tone embedding

A persistent misconception is that neural amp modeling is intrinsically one-model-per-preset. The recent literature does not support that view: parametric, latent, and one-to-many systems all explicitly target scalable deployment across many tones, settings, or devices, albeit with different trade-offs in interpretability and per-device accuracy (Chen et al., 2024, Grötschla et al., 30 Sep 2025).

2. Core modeling architectures for virtual analog amplification

Several architecture families dominate recent amp-emulation research. A feedforward WaveNet variant was used to emulate the Fender Bassman 56F-A vacuum-tube preamplifier with a 10-layer dilated causal convolution stack, filter width 3, dilation schedule $d_k=\{1,2,4,\ldots,512\}$, and receptive field $N=2046$ samples, approximately $46$ ms at $44.1$ kHz. Conditioning enters each layer through $1\times1$ convolutions, and training uses an error-to-signal ratio objective with the pre-emphasis filter $H(z)=1-0.95z^{-1}$. The larger configuration, WaveNet2, has about $30{,}000$ parameters, achieved ESR values between $0.32\%$ and $2.0\%$ across the tested gain and input-level conditions, received the highest mean MUSHRA scores, and ran at $3.0\times$ real time on a $N=2046$0 GHz Intel Core i5 CPU (Damskägg et al., 2018).

LSTM-based black-box emulation provides a different causal formulation. A real-time model of the ENGL Retro Tube 50 used raw waveform input at $N=2046$1 kHz, a 20-second dataset, and a single-layer LSTM with a dense output layer. For fixed-gain modeling, $N=2046$2 and $N=2046$3 achieved less than $N=2046$4 RMSE on validation data. For parametric Gain control, the input feature dimension was expanded to $N=2046$5, and a real-time configuration with $N=2046$6 and $N=2046$7 achieved less than $N=2046$8 RMSE. Real-time inference was demonstrated on an NVIDIA GTX 1050, with buffer carry-over preserving continuity in streaming operation (Schmitz et al., 2018).

A gray-box alternative is the DDSP amplifier model, which maps familiar circuit blocks into differentiable modules: preamp, tone stack, power amp, and output transformer. The preamp is modeled by $N=2046$9 cascaded Wiener–Hammerstein stages, the tone stack by a low-shelf/peak/high-shelf biquad cascade, the power amp by a push–pull topology with feedback-emulating filter $46$0, and the transformer by a GRU-based stateful nonlinearity plus bandpass-like filtering. On a Marshall JVM 410H (OD1 channel, high distortion), the full system achieved MAE $46$1 on seen settings and $46$2 on unseen settings, MR-STFT $46$3 and $46$4, and $46$5 ops/sample. The large black-box baseline still had the lowest losses, but required $46$6 ops/sample; the DDSP model therefore operated at less than $46$7 of that per-sample compute while preserving explicit control mappings (Yeh et al., 2024).

Hybrid engineering comparisons remain relevant. A comparative study of the Klon Centaur overdrive partitioned the circuit into input buffer, gain stage, tone control, and output buffer, then combined NA/MNA, WDF, and a compact RNN. The RNN used a single recurrent layer of $46$8 GRUs and was trained separately at five Gain settings. On a desktop plugin, the ML configuration processed one second of audio in $46$9–$44.1$0 seconds across block sizes $44.1$1–$44.1$2, compared with $44.1$3–$44.1$4 for the mixed non-ML model; the same codebase was also deployed on a Teensy 4.0 embedded pedal (Chowdhury, 2020).

Work	Model family	Reported outcome
“Deep Learning for Tube Amplifier Emulation” (Damskägg et al., 2018)	Feedforward WaveNet	ESR $44.1$5–$44.1$6; $44.1$7 real time on CPU
“Real Time Emulation of Parametric Guitar Tube Amplifier With Long Short Term Memory Neural Network” (Schmitz et al., 2018)	LSTM	$44.1$8 RMSE fixed Gain; $44.1$9 RMSE parametric Gain
“DDSP Guitar Amp: Interpretable Guitar Amplifier Modeling” (Yeh et al., 2024)	DDSP gray-box	MAE $1\times1$0; $1\times1$1 ops/sample
“A Comparison of Virtual Analog Modelling Techniques for Desktop and Embedded Implementations” (Chowdhury, 2020)	NA/WDF/RNN hybrid	ML runtime $1\times1$2–$1\times1$3 s/s audio

Taken together, these results show that the current architecture space is not polarized between “accurate but opaque” and “interpretable but weak.” Gray-box and hybrid systems occupy an intermediate regime, while black-box WaveNet and LSTM models remain competitive where maximal fidelity to a specific reference device is the primary objective.

3. One-to-many modeling, tone embeddings, and zero-shot control

A central recent development is the shift from one-to-one emulation toward one-to-many conditional rendering. In this setting, a single generator maps a clean DI guitar signal $1\times1$4 to a target wet signal $1\times1$5 under the control of a tone embedding $1\times1$6. The motivation is explicitly scalability: one-to-one training scales poorly for “Amplitube”-like ecosystems with hundreds of amp/cab/mic combinations and parameter states, and it cannot generalize to new amplifiers without retraining (Chen et al., 2024).

The tone-embedding framework uses a separate contrastive encoder trained on mel-spectrograms of wet guitar audio at $1\times1$7 kHz. Positives are clips with different musical content but the same tone; negatives have different tones. The encoder is trained with a SimCLR-style InfoNCE objective,

$1\times1$8

with cosine similarity and robustness augmentations including random cropping and additive noise. The resulting $1\times1$9-dimensional embedding is frozen during generator training (Chen et al., 2024).

The conditional generator itself is a time-domain gated CNN with residual connections and increasing dilations. It uses $H(z)=1-0.95z^{-1}$0 Conv1D layers, $H(z)=1-0.95z^{-1}$1 channels per layer, tanh/sigmoid gating, residual summation, and a final $H(z)=1-0.95z^{-1}$2 Conv1D mixing layer. Mono waveforms at $H(z)=1-0.95z^{-1}$3 kHz are processed in $H(z)=1-0.95z^{-1}$4-second segments. Conditioning is implemented by FiLM at every GCN layer: $H(z)=1-0.95z^{-1}$5 is projected to $H(z)=1-0.95z^{-1}$6 dimensions, then layer-specific MLPs generate $H(z)=1-0.95z^{-1}$7 so that

$H(z)=1-0.95z^{-1}$8

An ablation with simple concatenation was consistently worse, and FiLM outperformed both concatenation and LUT conditioning across amplifiers (Chen et al., 2024).

Training uses a complex STFT reconstruction loss with window $H(z)=1-0.95z^{-1}$9 and hop $30{,}000$0,

$30{,}000$1

with $30{,}000$2 dBFS peak normalization, Adam at $30{,}000$3, batch size $30{,}000$4, and convergence in about $30{,}000$5 days on an NVIDIA RTX 3090. The generator training set contains $30{,}000$6 minutes of monophonic DI, rendered through $30{,}000$7 amplifiers spanning three tone categories: high-gain, low-gain, and crunch. The encoder was trained on a larger in-house corpus with diverse amplifiers, pedals, and comprehensive settings (Chen et al., 2024).

Evaluation shows a consistent hierarchy. Per-amp one-to-one GCN baselines achieve the lowest losses and therefore define an upper bound. Among one-to-many variants, FiLM-GCN with tone embeddings substantially outperforms LUT conditioning and concatenation. Unpaired referencing—using a different-content reference clip $30{,}000$8 with the same tone as $30{,}000$9—slightly outperforms paired referencing, which was interpreted as evidence of stronger style/content disentanglement. Representative seen-amp results include $0.32\%$0 for amp4 versus LUT $0.32\%$1, and $0.32\%$2 for amp9 versus LUT $0.32\%$3. In zero-shot tests, unseen High Gain EL34 V2 yielded complex STFT losses of about $0.32\%$4 for direct conditioning, $0.32\%$5 for nearest retrieval, and $0.32\%$6 for mean-centroid retrieval; unseen Dumble ODS 50 yielded about $0.32\%$7, $0.32\%$8, and $0.32\%$9, respectively. The losses remained within $2.0\%$0–$2.0\%$1 of seen amps in the same category, and t-SNE visualizations showed two larger embedding clusters roughly separating high-gain and low-gain regimes (Chen et al., 2024).

This literature reframes tone matching as a representation-learning problem rather than a catalog lookup. It also makes explicit that match-to-reference workflows, zero-shot cloning of unseen amplifiers, and continuous interpolation in a shared “tone space” are technically connected rather than separate application classes.

4. Parametric control, active learning, synthetic corpora, and latent tone spaces

Continuous knob control has been addressed most directly by PANAMA. In the extended version, Panama trains a parametric model conditioned on a knob vector $2.0\%$2 containing Gain, Bass, Mid, Treble, Master, and Presence. The acquisition loop uses a $2.0\%$3-member LSTM ensemble for uncertainty estimation and a WaveNet-like final model for deployment. Disagreement combines waveform and Mel-domain variances across ensemble outputs, and gradient-based optimization with Adam maximizes this disagreement over the continuous knob space to propose informative settings. The practical loop starts from $2.0\%$4 random settings, uses $2.0\%$5 restarts per round, clusters the candidates, keeps $2.0\%$6–$2.0\%$7 unique settings per round, and stops after about $2.0\%$8 rounds, for roughly $2.0\%$9 datapoints (Grötschla et al., 30 Sep 2025).

The reported efficiency gains are substantial. Active learning achieved test MSE $3.0\times$0 and Mel loss $3.0\times$1, compared with $3.0\times$2 and $3.0\times$3 for random sampling and $3.0\times$4 and $3.0\times$5 for an extreme-value beta heuristic. In MUSHRA tests, Ours-75 showed no significant difference from NAM, and both were comparable to the reference. The LSTM ensemble trained at $3.0\times$6M samples/s on an RTX 3090, compared with $3.0\times$7M for WaveNet training, which is why the ensemble is used during acquisition and the WaveNet during final training (Grötschla et al., 30 Sep 2025). The earlier PANAMA abstract already showed the same tendency in a data-constrained regime: under a budget of $3.0\times$8 datapoints, active learning reached validation MSE $3.0\times$9, versus $N=2046$00 for uniform random sampling and $N=2046$01 for $N=2046$02 sampling (Grötschla et al., 2 Jul 2025).

A separate route to scale is synthetic data. Open-Amp crowdsources neural captures from GuitarML and Neural Amp Modeler ecosystems and renders them online during training. The reported corpus uses $N=2046$03 amp captures and $N=2046$04 pedal captures, $N=2046$05 in total, with $N=2046$06 exposing a single control; all supplied captures are single-layer LSTM models with hidden size $N=2046$07. For one-to-many training, controllable devices were discretized into five settings, expanding the synthetic device count to $N=2046$08. A $N=2046$09-minute clean corpus was then rendered into about $N=2046$10 hours of synthetic training material (Wright et al., 2024).

Open-Amp supports two distinct model classes. Its contrastive encoder uses a $N=2046$11D CNN with six residual blocks and produces a $N=2046$12-dimensional embedding from global average pooling; it has $N=2046$13 parameters. On guitar-effects classification, the Open-Amp encoder plus a $N=2046$14-layer MLP reached $N=2046$15 average accuracy across GFX splits, compared with $N=2046$16 for FxNet. Its one-to-many foundation model is a TCN with FiLM conditioning and learned device embeddings of size $N=2046$17. Emb-256 was generally closest to one-to-one TCN baselines; for a representative best device, one-to-one ESR was $N=2046$18 dB and Emb-256 reached $N=2046$19 dB with similar MRSL. Unseen analog pedals were enrolled by freezing the TCN and learning only a new device embedding, a few-shot adaptation procedure that narrowed or occasionally reversed the gap to one-to-one training under data scarcity (Wright et al., 2024).

Unpaired training addresses the data-collection bottleneck from another angle. A GAN-based clean-to-rendered transformer replaced MelGAN’s MSD-only discriminator with MSD+MPD, removed the lowest-resolution MSD path, and mixed extra unaligned clean audio into training. On the high-gain BD-2 target, supervised training gave L1mel $N=2046$20, ESR $N=2046$21, and FAD $N=2046$22; MSD+MPD reduced these to $N=2046$23, $N=2046$24, and $N=2046$25; MSD+MPD with both clean datasets further improved them to $N=2046$26, $N=2046$27, and $N=2046$28. For lower-gain EGDB targets, supervised ESR often remained best, but MSD+MPD consistently improved over MSD alone and often reduced FAD, indicating better distributional plausibility (Chen et al., 2024).

Latent-space design provides yet another control regime. A six-block convolutional architecture with fixed logarithmically spaced bandpass FIR filters, positive EQ weights $N=2046$29, asymmetric bias, softsign nonlinearity, and residual gating was trained on $N=2046$30 minutes of DI and six amplifier styles. Because every model shares the same fixed filter basis, linear interpolation and extrapolation over parameters $N=2046$31 permit explicit timbral arithmetic, such as blending Fender Twin, Vox AC30, Marshall Super Lead, Marshall JCM800, Mesa Boogie Triple Rectifier, and Peavey 5150 characteristics (Taylor, 2020).

Across these systems, the central trend is clear: “Amplitube-like” behavior is increasingly achieved through learned condition spaces rather than exhaustive preset libraries. Whether the condition variable is a knob vector, a tone embedding, a device embedding, or a latent parameter vector, the objective is the same—continuous control over a large family of tones with reduced capture burden.

5. Distortion physics, aliasing, and perceptual constraints

The signal-theoretic basis of virtual amplifier modeling remains the nonlinear generation of new spectral components. A polynomial nonlinearity

$N=2046$32

applied to $N=2046$33 generates harmonics and intermodulation products. The second-order term yields $N=2046$34; the third-order term yields reinforced fundamentals, third harmonics, and $N=2046$35, $N=2046$36. For a just power fifth with $N=2046$37, both quadratic and cubic distortion generate $N=2046$38, a sub-octave below the root; for a just major third with $N=2046$39, quadratic distortion produces $N=2046$40 and cubic terms generate a perfect fifth at $N=2046$41. The same analysis explains why equal temperament becomes fragile under high distortion: the just major third is $N=2046$42 cents, whereas equal temperament places it at $N=2046$43 cents, an error of about $N=2046$44 cents; the minor seventh differs by about $N=2046$45 cents (Mullin et al., 7 Apr 2025).

For virtual amp systems, another constraint is discrete-time aliasing. Neural models create harmonics at every nonlinear activation layer, and any generated energy above Nyquist folds back into the baseband. “Aliasing Reduction in Neural Amp Modeling by Smoothing Activations” formalizes this with the Aliasing-to-Signal Ratio,

$N=2046$46

measured from a prime-length DFT with $N=2046$47 and no windowing. Across $N=2046$48 trained models, smoother activations consistently lowered ASR. The lowest average ASR was reported for False_CustomTanh_32, at $N=2046$49, with ESR $N=2046$50; the best balanced tanh sweep occurred near $N=2046$51, with ASR $N=2046$52 and ESR $N=2046$53. Gating tended to improve ESR but increase aliasing substantially, while activation smoothing reduced aliasing without oversampling or post-filtering overhead (Sato et al., 7 May 2025).

A plausible implication is that audible realism in “Amplitube-like” systems is now constrained by three coupled objectives rather than one: low waveform or spectral error, low aliasing, and control fidelity under parameter changes. This is why the current literature increasingly mixes time-domain losses, multi-resolution spectral losses, perceptual or adversarial discriminators, and architectural priors such as gray-box DSP blocks or smooth nonlinearities (Yeh et al., 2024, Chen et al., 2024, Sato et al., 7 May 2025).

6. Amplitubes in graph-based cosmology and positive geometry

In mathematical physics, “amplitubes” are not audio devices but amplitude-like functions associated to graphs. For a graph $N=2046$54, a tube is a connected induced subgraph, and a tubing is a compatible collection of tubes. Graph associahedra generalize the classical associahedron from path graphs to arbitrary graphs, while graph cosmohedra are obtained by consistently blowing up all boundaries of the corresponding graph associahedron to codimension one. Path graphs recover the usual associahedra and cosmohedra; empty and complete graphs recover the simplex and permutohedron on the associahedron side, and the permutohedron and permutoassociahedron on the cosmohedron side (Glew et al., 24 Feb 2025).

In the graph-associahedral presentation, the amplitube is the vertex sum

$N=2046$55

where $N=2046$56 denotes maximal tubings and $N=2046$57 are linear functions of kinematic variables. The associated cosmological amplitube is

$N=2046$58

with $N=2046$59 maximal regional tubings and $N=2046$60 region variables. Facet factorization is built into the geometry: residues of $N=2046$61 factorize into products of lower amplitubes on $N=2046$62 and the reconnected complement $N=2046$63, while boundaries of $N=2046$64 factorize into an amplitube on a spine graph times cosmological amplitubes on regional reconnected components (Glew et al., 24 Feb 2025).

A complementary formulation distinguishes unary and binary tubes. Unary tubes are connected vertex subsets; binary tubes are connected subgraphs specified by vertices and edges. In this language,

$N=2046$65

while the flat-space wavefunction coefficient admits the cut expansion

$N=2046$66

This expresses the wavefunction coefficient for a connected graph as a sum over products of amplitubes of the connected components that remain after cutting subsets of edges. The same paper introduces cut tubings and decorated orientations, with the set of acyclic decorated orientations counting the number of basis functions appearing in the kinematic flow (Glew, 17 Mar 2025).

The correlator-level refinement is especially notable. For conformally coupled scalars in an FRW setting, amplitubes reorganize not only wavefunction coefficients but full correlation functions. The correlator expansion has the form

$N=2046$67

where $N=2046$68 if the contracted graph $N=2046$69 is bipartite and $N=2046$70 otherwise. The consequence is a hidden simplification relative to the wavefunction: at tree level, no term vanishes and the relative minus signs disappear; at loop level, many terms vanish because the corresponding contracted graphs are not bipartite. In the two-cycle example, the wavefunction has four amplitube terms with alternating signs, whereas the correlator keeps only two positive terms; in the sunrise example, eight wavefunction terms reduce to two correlator terms (Glew, 9 Jul 2025).

The word “amplitubes” therefore names a precise positive-geometric structure in cosmology: a sum over maximal tubings of graph associahedra, extended by graph cosmohedra, cut expansions, and bipartite-contraction rules for correlators. This usage is mathematically unrelated to virtual guitar amplification, but the shared name has become established in both literatures.