EQPred Model Architecture

• EQPred Model Architecture is a hybrid framework combining recurrent, implicit, and neuromorphic models to perform advanced signal processing and inverse problems.
• It employs LSTM equalizers, convolutional autoencoders with TCNs, and deep equilibrium modules to achieve resilient spatiotemporal predictions.
• The design integrates equilibrium-based parameter estimation with hardware-optimized implementations, yielding superior performance across diverse applications.

The EQPred model architecture refers to a set of hybrid, recurrent, implicit, and neuromorphic deep learning models developed for advanced signal processing, inverse problems, and spatiotemporal prediction. While several research lines reference the abbreviation "EQPred," common core themes include equilibrium-based parameter estimation, deep equilibrium modeling, neuromorphic implementations, and energy-based computational primitives. The following sections synthesize the primary architectural instantiations of EQPred as represented in peer-reviewed preprints.

1. Core Architectural Motifs and Mathematical Foundations

EQPred architectures are characterized by their use of deep recurrent or implicit neural modules built to robustly process complex, temporally and spatially structured data. Key instantiations employ Long Short-Term Memory (LSTM) networks with explicit gating mechanisms for signal equalization (Wang et al., 2020), convolutional autoencoder and temporal convolutional networks for spatiotemporal event nowcasting (Feng et al., 2020), and deep equilibrium (DEQ) or energy-based (EBM) modules for implicit deterministic inference (Hu et al., 2022, Tsuchida et al., 2022, Nest et al., 5 Sep 2024).

Fundamental mathematical relationships central to these models include:

• LSTM gating equations:

$$\begin{align*} f_t &= \sigma(W_f x_t + W_{rf} h_{t-1} + b_f) \ i_t &= \sigma(W_i x_t + W_{ri} h_{t-1} + b_i) \ \text{cs}_t &= \tanh(W_c x_t + W_{rc} h_{t-1} + b_c) \ o_t &= \sigma(W_o x_t + W_{ro} h_{t-1} + b_o) \ c_t &= f_t \odot c_{t-1} + i_t \odot \text{cs}_t \ h_t &= o_t \odot \tanh(c_t) \end{align*}$$

where $\sigma(\cdot)$ is the sigmoid function, $\tanh(\cdot)$ is the hyperbolic tangent, and $\odot$ is element-wise multiplication (Wang et al., 2020).

• Autoencoder-based spatial encoding:

$S_{i,j} = \sum_m \sum_n X_{i+m, j+n} \cdot K(m,n)$

• Deep equilibrium (DEQ) fixed point:

$z^* = f(z^*; \theta)$

and in the context of estimation:

$$z^* = \arg\min_z \{ -\log p(x|g(z;W)) - \log p(z) \}$$

• Feedforward-tied EBM (ff-EBM) energy minimization:

$$E^k(s^k, \theta^k, x^k) = G^k(s^k) - {s^k}^T x^k + U^k(s^k, \theta^k)$$

and corresponding hybrid training gradients via nudged equilibrium propagation:

$$g_{\theta^k} = \frac{1}{2\beta}\left[\nabla_2 E^k(s^k_\beta, \theta^k, x^k) - \nabla_2 E^k(s^k_{-\beta}, \theta^k, x^k)\right]$$

(Nest et al., 5 Sep 2024).

2. Architectural Variants

Principal EQPred architectural forms include:

Variant	Key Building Blocks	Application Domain
LSTM-based Equalizer	LSTM w/ gates, FC decoder, FIR post-filter	Channel equalization (Wang et al., 2020)
ConvAutoencoder + TCN Predictor	Encoder-decoder with skip, TCN with attention	Earthquake nowcasting (Feng et al., 2020)
DEQ/AR Implicit Prior	CNN prior as fixed-point solver	Imaging inverse problems (Hu et al., 2022)
ff-EBM Hybrid Analog/Digital	Digital feedforward, analog energy-based blocks	Neuromorphic/classical AI (Nest et al., 5 Sep 2024)

LSTM-based EQPred models use recurrent gating to remember long-term dependencies in noisy channel signals and replace feedforward/decision-feedback structures with a unified plasticist equalizer (Wang et al., 2020). Spatiotemporal versions mine grid-structured images over time and predict future event likelihood through compression and temporal pattern extraction (Feng et al., 2020). Deep equilibrium formalisms (e.g., AR-priors) leverage implicit differentiation to enforce global optimality/consistency and can surpass plug-and-play denoisers in robustness to measurement model mismatch (Hu et al., 2022). New ff-EBM architectures use digital/analog block interleaving and specialized end-to-end gradient chaining for heterogeneous accelerator hardware (Nest et al., 5 Sep 2024).

3. Signal Processing and Spatiotemporal Modeling Mechanisms

The architectural details are tailored to domain-specific data statistics:

• Equalization: The LSTM-based EQPred receives a re-sampled signal buffer, sequences it through a recurrent cell, and produces equalized outputs via a fully connected decoder followed by a post-processing FIR filter. The design enables direct nonlinear interference suppression without manual filter tuning, and maintains adaptability across frequencies (10–60 GHz) by adjusting sequence/memory parameters (Wang et al., 2020).
• Spatiotemporal Mining: Earthquake prediction models partition geography into a grid and compute 2D “energy images” for each interval, using $E = (10^{\text{Mag}})^{3/2}$ as the cell value. The convolutional autoencoder serves as a low-dimensional spatial compressor, while the TCN with local attention dynamically weights the temporal evolution of features (Feng et al., 2020).

4. Training and Optimization Strategies

Training protocols and loss functions are problem-specific but universally exploit supervised or semi-supervised learning, stochastic optimization, and explicit or implicit differentiation.

• Supervised LSTM Equalizer: Offline mini-batch training via stochastic gradient methods (often Adam), targeting MSE loss:

$$\text{MSE} = \frac{1}{N} \sum_{i=1}^N (s_i - s'_i)^2$$

(Wang et al., 2020).

• Spatiotemporal AE+TCN: Joint optimization over spatial autoencoder reconstruction loss (e.g., MAE: $L = \text{MAE}(X - \hat{X})$) and temporal prediction loss (e.g., Nash–Sutcliffe efficiency for sequence matching) (Feng et al., 2020).
• DEQ/EBM-based Priors: Solve for implicit equilibrium states parameterized by task-specific measurement operators; gradient computation proceeds via implicit differentiation, as in

$$\nabla \ell(f_\theta) = \left(\nabla_x T(f_\theta, x^*)\right)^\top \left[-(\nabla_x T(f_\theta, x^*))^{-1}\right] \cdot (-(x^* - x))$$

(Hu et al., 2022). Energy-based blocks in ff-EBMs are trained using finite-difference estimators derived from equilibrium perturbations (Nest et al., 5 Sep 2024).

5. Robustness, Adaptability, and Hardware Realizations

EQPred models prioritize adaptability and support for advanced hardware implementations:

• Robustness to Channel and Model Mismatch: LSTM-based and equilibrium AR-prior models demonstrate robustness to shifting channel characteristics (frequency, impulse response, or measurement model). DEQ-based architectures trained on one measurement operator can outperform conventional denoisers even when evaluated on mismatched models—e.g., achieving a PSNR improvement of ~1.84 dB in CS-MRI reconstruction relative to AWGN denoising (Hu et al., 2022).
• Neuromorphic and Heterogeneous Hardware Integration: These architectures are specifically implemented for cross-platform compatibility, including digital (FPGA), analog, and hybrid neuromorphic devices. The LSTM equalizer cell is constructed for both FPGA and analog circuit realization, enabling operation at frequencies inaccessible to digital-only systems (Wang et al., 2020). Feedforward-tied EBM instantiations allow modular physical partitioning so that trainable analog blocks (e.g., as in-memory crossbar arrays) can be chained with digital accelerators via principled gradient propagation (Nest et al., 5 Sep 2024).

6. Empirical Performance and Comparative Evaluation

EQPred-based designs consistently surpass traditional signal processing and learning baselines. For channel equalization, LSTM-based EQPreds achieve superior Bit Error Rate (BER) and improved eye diagram metrics compared to decision-feedback equalizers, maintaining regularized waveform restoration across diverse frequency bands (Wang et al., 2020). In nowcasting, AE+TCN architectures obtain lower error and higher F-scores than MLP, LSTM, or simpler convolutional/recurrent baselines in extreme earthquake prediction (Feng et al., 2020).

In imaging, DEQ priors trained using artifact removal maintain higher PSNR ($36.76$ dB vs. $34.92$ dB for AWGN prior in CS-MRI) under measurement model mismatch, opposing earlier heuristic beliefs about the inherent superiority of plug-and-play robustness (Hu et al., 2022). For hybrid analog-digital models, ff-EBMs with Deep Hopfield Network blocks achieve state-of-the-art performance for equilibrium propagation (46% top-1 accuracy on ImageNet32) and exhibit empirical gradient match to implicit differentiation, confirming the viability of scalable, hybrid training (Nest et al., 5 Sep 2024).

7. Implications and Future Directions

The broad EQPred design space enables the integration of deep learning, signal processing, equilibrium optimization, and neuromorphic hardware into unified frameworks. The architectural advances allow adaptation to diverse data distributions and hardware constraints, eliminate handcrafted filter heuristics, and provide a theoretically grounded bridge between energy-based systems and digital AI pipelines.

A plausible implication is that continued research in hybrid equilibrium-feedforward models and BP–EP gradient chaining will facilitate further replacement of digital-centric AI with physically adaptive, highly energy-efficient equivalents, especially as analog and in-memory compute devices mature. The EQPred approach thus represents an intersection where model plasticity, interpretability, and cross-domain robustness are compatible with next-generation hardware and challenging application domains.