# Q-DEQ: Discrete Solving and Quantization for Deep Equilibrium Models in Time Series Forecasting under Edge Deployment Coding Constraints

arXiv: https://arxiv.org/abs/2609.24042 · HTML: https://mlsanity.com/paper/2609.24042/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 15 references that are in the map.
- First posted on arXiv 21 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference](https://mlsanity.com/paper/1712.05877/) (2017)
- [iTransformer: Inverted Transformers Are Effective for Time Series Forecasting](https://mlsanity.com/paper/2310.06625/) (2023)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Deep Graph Matching via Blackbox Differentiation of Combinatorial Solvers](https://mlsanity.com/paper/2003.11657/) (2020)
- [Deep Equilibrium Optical Flow Estimation](https://mlsanity.com/paper/2204.08442/) (2022)
- [LoopQ: Quantization for Looped Language Models](https://mlsanity.com/paper/2605.16343/) (2026)
- [Automating Versatile Time-Series Analysis with Tiny Transformers on Embedded FPGAs](https://mlsanity.com/paper/2505.17662/) (2025)
- [What Survives When You Compress a Recursive Reasoner for the Edge?](https://mlsanity.com/paper/2606.26488/) (2026)
- [Quantizing Recursive Reasoning Models](https://mlsanity.com/paper/2607.16237/) (2026)

## Abstract (arXiv)

Edge deployment motivates forecasting models with compact parameter storage and low-bit representations. Deep equilibrium models (DEQs) obtain implicit depth by repeatedly applying a shared layer, reducing the parameter cost of explicit layer stacking. Their usual Anderson solver, however, searches for update coefficients in the continuous real domain. We propose Q-DEQ, which formulates local updates in DEQ forward solving as discrete optimization problems. Candidate directions are constructed from the current state and iteration history, and a local quadratic residual model is used to evaluate their combinations. Binary encoding of the direction coefficients yields a quadratic unconstrained binary optimization (QUBO) problem that can be solved by simulated annealing (SA) or a coherent Ising machine (CIM). After fixed-point solving, a re-forward pass applies W8A8 fake quantization to the shared layer's weights and activations. We evaluate Q-DEQ with an iTransformer backbone on five multivariate time series forecasting datasets. Relative MSE differences from the explicit multi-layer baseline range from -1.16% to +2.90%, with lower MSE on two datasets. DEQ parameter sharing reduces parameter counts by factors of 1.80×--3.82×; combined with W8A8, static weight storage is reduced by factors of 4.3×--12.8×. Local QUBO problems solved using CPU-based SA and the Kaiwu CIM physical backend produce closely matching downstream forecasts. These results establish local discrete solving as a viable component of DEQ time series forecasting and provide a route for executing fixed-point updates through different combinatorial optimization backends.