Differentiable Auditory Loop (DAL)

• Differentiable Auditory Loop (DAL) is a framework that integrates recursive audio processing, perceptual models, and gradient-based loss optimization for end-to-end learning.
• It encompasses multiple forms—from differentiable recursive DSP in FDNs to auditory-model guided hearing-aid optimization—ensuring stability and fidelity via parameter constraints and certification.
• Recent advances like ADAC enable real-time, audible optimization by hot-swapping trained differentiable graphs into live DSP plugins with minimal latency.

Searching arXiv for the cited DAL-related papers and topic context. The Differentiable Auditory Loop (DAL) is an end-to-end differentiable framework in which an audio processor, an auditory or acoustic model, and an objective are coupled so that gradients can propagate through the entire loop and update processor parameters by gradient-based optimization. In the literature, DAL appears in at least three closely related forms: as a differentiable recursive audio processor such as a Feedback Delay Network (FDN), as a hearing-aid optimization loop in which a differentiable auditory periphery defines the loss domain, and as a broader closed-loop predictive system in which prediction errors are fed back to update latent states. A distinctive recent development is that DAL is no longer confined to offline training: ADAC makes the optimization loop audible by recompiling and hot-swapping a trained differentiable audio graph into a running real-time plugin after each optimizer step (Mezza et al., 2024, Gonzalez et al., 12 Jul 2025, Rosen et al., 2 Jun 2026, Franchino et al., 19 Jun 2026).

1. Terminology and conceptual scope

One explicit definition describes DAL as “an audio feedback loop whose parameters (e.g., delays, gains, and mixing/feedback matrices) are optimized end-to-end via backpropagation and automatic differentiation against perceptual objectives,” with all loop components exposed to gradient flow by making their signal operations differentiable (Mezza et al., 2024). A second formulation defines DAL as placing “a differentiable computational auditory model inside the training loop so that gradients can propagate from a perceptual-domain objective back to the signal-processing network” (Gonzalez et al., 12 Jul 2025). In hearing-aid work, this usually means comparing an impaired auditory pathway driven by processed audio against a normal-hearing pathway driven by a reference signal, then backpropagating through the impaired model to optimize the processor (Tu et al., 2021, Drakopoulos et al., 2022, Rosen et al., 2 Jun 2026).

The term is not used identically in all papers. In the biomimetic frontend literature, the “loop” is not a physical feedback loop but a learning loop in which task losses drive parameter updates through a fully differentiable auditory pipeline (Famularo et al., 2024). In auditory working-memory modeling, the relevant architecture is introduced as a “close-loop predictive coding” framework rather than DAL, but it is described as functionally equivalent because predictions, errors, and state updates form a differentiable closed loop (Yuan et al., 16 Mar 2025). This suggests that DAL is best understood as a family of differentiable closed-loop auditory systems rather than a single fixed architecture.

A compact way to organize the literature is the following.

Formulation	Representative papers	Characteristic loop
Differentiable recursive DSP	(Mezza et al., 2024, Franchino et al., 19 Jun 2026, Gerami et al., 30 Sep 2025)	Trainable delays, gains, filters, and feedback
Differentiable auditory supervision	(Tu et al., 2021, Drakopoulos et al., 2022, Gonzalez et al., 12 Jul 2025, Rosen et al., 2 Jun 2026)	Processor optimized through auditory-model losses
Closed-loop predictive coding	(Yuan et al., 16 Mar 2025)	Prediction error fed back to latent-state updates

2. Canonical computational pattern

Despite domain differences, DAL systems share a common computational structure. An input waveform is transformed by a parametric processor, the processor output is analyzed by a differentiable auditory or acoustic model, and a loss is computed against a reference representation. Optimization then follows standard gradient-based updates such as

$$\theta_{t+1}=\theta_t-\eta\nabla_\theta L(\theta_t),$$

with Adam used in several implementations (Franchino et al., 19 Jun 2026, Mezza et al., 2024).

In DHASP, the hearing-aid processor $P_\theta$ produces $y(t)=P_\theta(x)$, and the loss compares envelope-based representations from a normal-hearing path $A_{\mathrm{NH}}(x)$ and a hearing-impaired path $A_{\mathrm{HI}}(P_\theta(x))$. The training objective is implemented as

$$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$

where $R(j)$ is a normalized cepstral correlation and $L_e^{(i)}$ is an energy-control penalty (Tu et al., 2021). In the controllable joint NR+HLC framework, the processor outputs two waveforms, $\hat{y}_{\mathrm{NR}}$ and $\hat{y}_{\mathrm{HLC}}$, and losses are computed in the output space of a differentiable auditory model:

$P_\theta$0

$P_\theta$1

A trainable uncertainty-weighted multi-task objective combines these terms (Gonzalez et al., 12 Jul 2025).

In the CARFAC-based DAL, the normal-hearing path produces target Neural Activity Patterns (NAPs) and Stabilized Auditory Images (SAIs), while the impaired path passes the output of a causal SEANet through an impaired CARFAC model. A generic DAL loss is written as

$P_\theta$2

with $P_\theta$3 instantiated by SSIM- or PN-based distances (Rosen et al., 2 Jun 2026).

The same pattern also appears outside hearing aids. In the predictive-coding memory model, the hidden state is updated by minimizing a local free-energy-like objective,

$P_\theta$4

and the observation error during read is computed relative to the most recently recalled segment, creating an explicitly closed loop (Yuan et al., 16 Mar 2025).

Framework support reflects this generality. PyTorch is used in DHASP and differentiable FDN work; TensorFlow/Keras is used in CoNNear-based hearing-loss compensation; JAX/Flax is used in CARFAC-based DAL and in the biomimetic differentiable frontend (Tu et al., 2021, Mezza et al., 2024, Drakopoulos et al., 2022, Famularo et al., 2024, Rosen et al., 2 Jun 2026).

3. Differentiable recursion, delay lines, and stability

A central technical problem in DAL research is differentiability through recursive audio structures. The clearest examples arise in FDN-based reverberation and room-acoustic modeling. For an FDN with delay vector $P_\theta$5, feedback matrix $P_\theta$6, input/output gains $P_\theta$7, and direct path $P_\theta$8, the transfer function is

$P_\theta$9

with

$y(t)=P_\theta(x)$0

In differentiable FDN training, all parameters in $y(t)=P_\theta(x)$1 may be learned by backpropagation (Mezza et al., 2024, Franchino et al., 19 Jun 2026).

The most distinctive difficulty is the delay length. In the learnable-delay FDN, fractional delays are implemented in the frequency domain through a phase factor $y(t)=P_\theta(x)$2 applied to the FFT of a buffer, so that gradients can flow through the delay via

$y(t)=P_\theta(x)$3

This is combined with constrained parameterizations for stability and causality: nonnegative gains are obtained with $y(t)=P_\theta(x)$4, delays satisfy $y(t)=P_\theta(x)$5, and the feedback matrix is factorized as $y(t)=P_\theta(x)$6 with $y(t)=P_\theta(x)$7 orthogonal and $y(t)=P_\theta(x)$8 in $y(t)=P_\theta(x)$9 (Mezza et al., 2024).

ADAC addresses a different stage of the same problem: faithful lowering of a trained differentiable audio graph to real-time DSP. It extracts a framework-agnostic JSON intermediate representation with Series, Parallel, and Recursion nodes and emits FAUST code. Structural mapping is explicit: Series maps to :, Parallel to shared-input split, and Recursion to ~. Because FAUST recursion introduces one implicit sample delay, ADAC corrects loop timing by replacing each in-loop delay $A_{\mathrm{NH}}(x)$0 by $A_{\mathrm{NH}}(x)$1 and restoring absolute arrival time with an external $A_{\mathrm{NH}}(x)$2, using

$A_{\mathrm{NH}}(x)$3

For integer delays, the emitted graph matches the source sample-for-sample up to float arithmetic; across stereo FDNs and mono FDNs with direct paths, the residual is at or below $A_{\mathrm{NH}}(x)$4 of peak, and the magnitude-response difference is at or below $A_{\mathrm{NH}}(x)$5 dB of peak (Franchino et al., 19 Jun 2026).

Stability is treated as a first-class DAL concern. ADAC computes a small-gain certificate on shipped parameters,

$A_{\mathrm{NH}}(x)$6

and also runs pole checks on every recursive filter section. The certificate is written as JSON and checked before building; the exporter refuses to build a model whose certificate is unstable or unproven unless explicitly overridden (Franchino et al., 19 Jun 2026). In the metric-driven room-rendering FDN, stability is obtained by the specialization $A_{\mathrm{NH}}(x)$7 with $A_{\mathrm{NH}}(x)$8, which yields a parallel-loop DAL with $A_{\mathrm{NH}}(x)$9 complexity and reported cost of 63 FLOPs/sample for the tail and 149 FLOPs/sample including early reflections and their HRIR application (Gerami et al., 30 Sep 2025).

4. Audible optimization, compilation, and sound quality

DAL initially referred primarily to differentiable optimization, but recent work made the optimization process itself audible. ADAC operationalizes this by hot-swapping the processor into a running plugin after each optimizer step. After each step, ADAC re-emits FAUST .dsp code and atomically rewrites a watched file; a resident plugin built on the FAUST interpreter detects file changes and reloads or recompiles the DSP graph on the fly. Recompilation of the emitted network is under 10 ms on an Apple M2, ADAC’s re-emission time is approximately 0.2 ms, and audible changes occur within a few audio buffers after each gradient step. Publishing is deduplicated and rate-limited, atomic replacement avoids torn reads, and slider values survive reloads “by address” (Franchino et al., 19 Jun 2026).

The same system couples audibility with deployment. FAUST code can be compiled to audio plugins, web, embedded boards, and FPGA targets, including VST3, AU, CLAP, WebAssembly, Bela, Daisy, ESP32, and VHDL via Syfala. A single call executes code generation, certification, JUCE project generation, release compilation, and installation of validated binaries into plugin folders. Three macro-controls are shipped—reverberation time, dry/wet balance, and pre-delay—and the reverberation-time control uses Jot’s homogeneous decay,

$A_{\mathrm{HI}}(P_\theta(x))$0

which was validated in the compiled plugin: measured via Schroeder integration, $A_{\mathrm{HI}}(P_\theta(x))$1 s yields exactly 0.5000 s (Franchino et al., 19 Jun 2026).

Closed-loop sound quality has also motivated architectural redesign. The dCoNNear framework was introduced because earlier DNN-based closed-loop systems suffered from aliasing, tonal artifacts, and imaging artifacts caused by strided downsampling and transposed or subpixel upsampling. dCoNNear removes all downsampling and upsampling between layers and uses FIR-like memory blocks with dilated depthwise convolutions at a single sampling rate. In the reported artifact measurements, the hearing-aid model’s fractional-harmonic THD improved from $A_{\mathrm{HI}}(P_\theta(x))$2 dB for prior CoNNear to $A_{\mathrm{HI}}(P_\theta(x))$3 dB for dCoNNear, and SRMR on noisy speech improved from 5.17 to 6.24, although music SRMR remained slightly higher for prior CoNNear (Wen et al., 7 Jan 2025). This is important because DAL objectives can otherwise be spuriously improved by artifact-generating architectures rather than by genuinely better auditory processing.

5. Hearing-aid personalization and auditory-model supervision

The most extensive DAL literature concerns hearing aids and hearing-loss compensation. DHASP established an explicitly differentiable hearing-aid speech-processing pipeline in which a frequency-dependent linear amplifier, parameterized at six audiogram frequencies $A_{\mathrm{HI}}(P_\theta(x))$4 Hz, is optimized using a differentiable HASPI-inspired auditory model. The processor is realized as an FIR filter constructed by frequency-response interpolation and IFFT, and the objective is based on cepstral correlation between normal-hearing and hearing-impaired auditory envelopes plus an energy-control penalty. Training used Adam with learning rate 0.001, batch size 128, and 4000 epochs on 0.5 s TIMIT segments normalized to 65 dB SPL equivalent. Across 10 standard audiograms, the optimized filters achieved higher HASPI intelligibility than NAL-R, with improvements significant at paired $A_{\mathrm{HI}}(P_\theta(x))$5-test $A_{\mathrm{HI}}(P_\theta(x))$6 (Tu et al., 2021).

A more physiologically detailed DAL is built from CoNNear models of cochlea, inner hair cell, and auditory nerve. In that framework, a DNN-based hearing-aid processor $A_{\mathrm{HI}}(P_\theta(x))$7 is trained so that the hearing-impaired periphery response $A_{\mathrm{HI}}(P_\theta(x))$8 matches the normal-hearing response $A_{\mathrm{HI}}(P_\theta(x))$9 under losses on neurograms, population responses, and STFT magnitudes. The 16-layer encoder–decoder CNN has about 5,197,633 trainable parameters. Results show that outer-hair-cell loss is easier to compensate than cochlear synaptopathy: for the OHC-only Slope25 profile, the DNN-HA achieved average NRMSE below 1% at conversational levels, whereas CS-only restoration was partial; for the mixed Slope35-7,0,0 profile, the best EFR restoration increased the simulated response from 4.31 nV unprocessed to 7.75 nV, near the 8.39 nV normal-hearing reference (Drakopoulos et al., 2022).

The artifact-free dCoNNear closed-loop system takes a related but distinct route. It replaces non-DNN biophysical models with differentiable DNN surrogates of cochlea, IHC, and ANF processing, freezes those auditory modules, and trains a dCoNNear hearing-aid processor to minimize

$$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$0

with $$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$1 and $$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$2. Both prior CoNNear HA and dCoNNear HA reduce NRMSE between NH and HI AN population responses, but the paper argues that prior CoNNear’s slightly better NRMSE is partly artifact-driven, whereas dCoNNear provides better sound quality (Wen et al., 7 Jan 2025).

Later DAL systems integrate personalization, denoising, and controllability. The controllable joint NR+HLC model uses a BSRNN backbone conditioned on audiograms via FiLM and optimized against a differentiable simplified CASP model. It outputs $$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$3 and $$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$4 and exposes an inference-time mixing control

$$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$5

On NH evaluation at $$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$6, the controllable joint model reached SDR 13.00, PESQ 2.14, and ESTOI 0.85. On HI evaluation, $$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$7 gave HASPI 0.77, while $$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$8 gave the best HASQI at 0.61 (Gonzalez et al., 12 Jul 2025).

The CARFAC-based DAL pushes this trajectory further by using a subject-specific impaired cochlear model and a causal SEANet waveform-to-waveform UNet generator. CARFAC v3 was ported to JAX/Flax with 77 channels up to 14.4 kHz at 32 kHz sampling, and the proof-of-concept impairment was modeled by setting OHC health to 0.5, producing a mild but significant loss with more than 30 dB HL around 3–4 kHz. On NAP metrics, the noisy baseline had L1 0.099, correlation 0.428, and SI-SDR $$L(\theta;\phi)= -\frac{1}{5}\sum_{j=2}^{6}R(j)+\alpha\sum_i L_e^{(i)},$$9 dB, while SEANet trained with L1-NAP achieved L1 0.073, correlation 0.706, and SI-SDR $R(j)$0 dB. On SAI metrics, SEANet variants improved correlation to 0.853–0.870, outperforming the tested MHA baselines (Rosen et al., 2 Jun 2026).

6. Extensions, misconceptions, and open problems

DAL should not be reduced to hearing-aid fitting or to explicit time-domain recursion. The biomimetic frontend for differentiable audio processing is a strictly feedforward, vectorized auditory pipeline comprising a constant-Q cochlear filterbank, hair-cell-like compression, lateral inhibition, short-term integration, and cortical STRFs. The paper explicitly notes that the loop is a learning loop rather than a physical feedback loop. The frontend has about 212 learnable parameters and serves classification and enhancement tasks; in enhancement, the fully differentiable frontend achieved SI-SDR 8.31±0.23 in the Original condition and 16.2±0.19 in the NewNoise condition, with the best or near-best performance across conditions (Famularo et al., 2024).

Similarly, the closed-loop predictive-coding framework for auditory working memory does not introduce the term DAL explicitly, but it matches the functional description of a differentiable closed loop. It uses a hidden size of 1,600 neurons, processes 200 ms segments, retains the first 20 steps corresponding to 4 s for memory, writes for 100 epochs with $R(j)$1 inner inference iterations, and reads with $R(j)$2 iterations. On ESC-50, most classes showed median semantic similarity above 0.7; on LibriSpeech, median semantic similarity was above 0.8, and closed-loop feedback substantially improved waveform alignment and recognition consistency relative to recall without feedback (Yuan et al., 16 Mar 2025).

A common misconception is that differentiability alone guarantees equivalence, stability, or perceptual validity. The literature repeatedly qualifies such claims. ADAC’s equivalence and stability guarantees are tied to the linear setting, even though FAUST can express nonlinear and time-varying operators (Franchino et al., 19 Jun 2026). The differentiable FDN room-acoustics model is time-invariant and frequency-independent, and the paper states that none of the tested models matched the target magnitude response well (Mezza et al., 2024). The controllable NR+HLC model removes outer-ear and modulation-filterbank stages from CASP for simplicity and runtime (Gonzalez et al., 12 Jul 2025). The CARFAC-based DAL currently reports only simulated OHC loss in experiments, while broader etiologies, clinical listening tests, and deployment measurements remain future work (Rosen et al., 2 Jun 2026).

Open problems are correspondingly clear. Extending guarantees from LTI graphs to nonlinear or time-varying processors, incorporating fractional-delay implementations rather than emission-time rounding, adding oversampling or ADAA-like approaches for nonlinear blocks, broadening impairment models beyond OHC loss, and validating auditory-domain objectives against human studies are all presented as future work rather than completed results (Franchino et al., 19 Jun 2026, Mezza et al., 2024, Gonzalez et al., 12 Jul 2025, Rosen et al., 2 Jun 2026). The current literature therefore establishes DAL less as a finished standard than as a differentiable systems framework whose concrete realization depends on how recursion, auditory modeling, stability, and deployment are jointly handled.