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MoFE: A Novel Mixture-of-Experts Framework with Fourier Neural Operators for Cryptocurrency Forecasting

Bowen Liu, Mingming Sun

arXiv:2608.17342v1cs.LGcs.AI

TL;DR

Cryptocurrency forecasting is challenged by phase lag, high-frequency noise, and rapid regime shifts. MoFE combines dual-domain Fourier Neural Operators with a dynamically gated Mixture-of-Experts architecture, achieving state-of-the-art performance for T+1 and T+5 forecasts while improving directional predictive power and simulated-trading returns.

  • Problem

    Existing cryptocurrency forecasting models often suffer phase lag, weak adaptation to rapid regime shifts, and sensitivity to high-frequency market noise.

  • Method

    MoFE combines dual-domain AFNO and convolutional experts with a dynamically gated Mixture-of-Experts framework to model global spectral trends, local microstructures, and market-regime changes.

  • Results

    MoFE achieves state-of-the-art results for T+1 and T+5 horizons, significantly reducing phase lag and enhancing directional predictive power.

  • Takeaways & Limitations

    In simulated trading, MoFE delivers exceptional risk-adjusted returns, indicating practical utility for cryptocurrency forecasting.

Abstract

from arXiv · show

Forecasting cryptocurrency prices remains a formidable challenge due to inherent non-stationarity, abrupt regime shifts, and multi-scale stochastic dependencies. Conventional deep learning models often struggle to capture complex underlying dynamics, frequently resulting in persistent phase-lagged predictions. To address these limitations, we propose MoFE, a novel deep learning framework that integrates Fourier Neural Operators (FNOs) within a Mixture-of-Experts (MoE) architecture. Rooted in the theoretical framework of stochastic differential equations, MoFE conceptualizes cryptocurrency volatility as a superposition of multi-frequency components, which includes user network based fundamental growth, mining costs and halving mechanism caused seasonal volatility, and market sentiment-induced chaos. Specifically, specialized adaptive FNO (AFNO) and Convolution dual-domain experts learn continuous function-to-function mappings to encapsulate global spectral trends, cyclical adjustments and microstructures, while a dynamic gating based MoE mechanism enables adaptive strategy switching across diverse market regimes. Extensive experiments on Bitcoin datasets spanning January 2020 to December 2025 demonstrate that MoFE achieves state-of-the-art (SOTA) performance in both T+1 and T+5 forecasting horizons. Notably, the model effectively mitigates the phase-lag effect, delivering superior Directional Accuracy (DA) and Information Coefficient (IC). In high-fidelity simulated trading environments, these predictive gains transfer into significant excess returns and robust risk-adjusted performance, characterized by a high Sharpe ratio.

I. INTRODUCTION

Cryptocurrency forecasting is challenged by sentiment-driven volatility, non-stationarity, high-frequency noise, and abrupt regime shifts that induce phase-lagged predictions. MoFE addresses these limitations through a dual-domain Mixture-of-Experts framework combining AFNO frequency modeling with convolutional temporal modeling.

  • Motivation: Cryptocurrency prices reflect network-driven fundamental growth, mining-cost and halving seasonality, and sentiment-induced chaos, without intrinsic valuation anchors.These dynamics can invalidate traditional mean-reversion frameworks.
  • Limitations: Conventional deep learning models often produce phase lags because point-wise regression fails to capture continuous spectral dynamics and rapid regime shifts.Time-domain-only models are also sensitive to high-frequency cryptocurrency market noise.
  • Method: MoFE combines frequency-domain and time-domain modeling, using global FNOs for long-range cyclical dynamics and convolutional branches for short-term temporal microstructures.The framework is designed to model non-stationary volatility and abrupt regime shifts through multi-expert decomposition and nonlinear optimization.
  • Method: The AFNO-Conv dual-domain operator captures global cyclical patterns in the frequency domain and transient local microstructures in the temporal domain.This design targets the multi-scale nature of financial data.
  • Results: MoFE achieves state-of-the-art performance across RMSE, R2, Sharpe Ratio, Information Coefficient, and ROI, with back-testing showing significant risk-adjusted returns and robust predictive consistency.The reported evaluation spans statistical metrics and financial indicators.

II. RELATED WORK · III. METHODOLOGY · A. Architecture Overview

The paper motivates cryptocurrency forecasting as difficult because Bitcoin exhibits extreme volatility, structural instability, non-linearity, and diverse market influences. MoFE responds by combining frequency-domain modeling with Mixture-of-Experts and a time-frequency dual-domain architecture built around parallel experts and gating.

  • II. RELATED WORK: Bitcoin valuation reflects market liquidity, mining costs, macroeconomic changes, and investor sentiment, making its behavior highly volatile and structurally unstable.These characteristics distinguish Bitcoin from traditional assets and complicate forecasting.
  • II. RELATED WORK: Bitcoin price prediction is exceptionally difficult because of inherent non-linearity, motivating robust forecasting frameworks for data-driven decisions.The stated motivation includes improving market transparency and supporting institutional investors and regulators.
  • II. RELATED WORK: Statistical approaches such as ARIMA and GARCH rely on linear dynamical systems, stochastic processes, stationarity, and historical autocorrelation.The passage characterizes these methods as constrained to local or univariate features.
  • II. RELATED WORK: Deep learning methods, including LSTM and GRU variants, use non-linear optimization and gating mechanisms to model high-dimensional financial dependencies and temporal dynamics.The passage specifically notes that GRU improves computational efficiency by consolidating part of the LSTM mechanism.
  • II. RELATED WORK: Frequency-domain methods exploit time-frequency duality, while Mixture-of-Experts frameworks dynamically coordinate specialized experts for financial prediction.FreqMoE demonstrates FNO–MoE integration for complex PDE systems, motivating their synergistic application here.
  • III. METHODOLOGY · A. Architecture Overview: MoFE begins with a Linear-layer Stem Block that projects the input sequence into a high-dimensional latent space with dmodel.This establishes the representation passed to the subsequent MoE block.
  • III. METHODOLOGY · A. Architecture Overview: The latent features are concurrently processed by K parallel Experts and a Gating Network within the MoE block.The architecture therefore combines multiple specialized processing paths with learned coordination.
  • A. Architecture Overview: Each MoFE expert is a time-frequency dual-domain feature extraction module whose global FNO path uses an adaptive FNO to capture global cycles.The supplied architecture description introduces the global FNO path but ends before fully specifying the remaining expert components.

B. Our MoFE Model · 1) FNO-Expert:

MoFE experts use dual paths to capture global periodic patterns alongside local transient microstructures. The FNO-expert filters frequency-domain noise, models localized market dynamics, and fuses both representations into the model dimension.

  • B. Our MoFE Model: Each MoFE expert uses a dual-path architecture to capture global periodic patterns and local transient microstructures.The expert maps hidden representations X ∈ R^B×L×C to outputs E_k(X) ∈ R^B×K×C.
  • 1) FNO-Expert:: Together, the dual paths synergistically combine frequency-domain periodic structure with spatially localized market behavior.This summarizes the expert’s complementary global and local modeling design.
  • 1) FNO-Expert:: The global FNO path uses a one-dimensional Adaptive Fourier Neural Operator to model long-range dependencies and cyclical market fluctuations.Its formulation uses FFT, inverse FFT, Layer-Norm, and a complex-valued MLP block.
  • 1) FNO-Expert:: The low-pass filter reduces spectral noise and redundancy through hard thresholding and soft shrinkage of Fourier coefficients.Soft shrinkage uses SS(Z) = sign(Z) max(0, |Z|−λ), where λ is a sparsity threshold.
  • 1) FNO-Expert:: The local context path uses a 1D convolution with kernel size n = 3 followed by batch normalization.This branch captures short-term momentum, abrupt shocks, and localized trend variations.
  • 1) FNO-Expert:: Global spectral features and local spatial contexts are integrated through element-wise summation.The fused representation is projected back to the model dimension using a 1 × 1 point-wise convolution and LeakyReLU.

2) Mixture-of-Experts System: · 3) Prediction Head:

MoFE uses a gating network to select and aggregate experts according to changing cryptocurrency market regimes. Its prediction head decodes aggregated latent features into forecasts through an MLP with nonlinear activation and stochastic regularization.

  • 2) Mixture-of-Experts System:: The MoE module addresses cryptocurrency markets’ high non-stationarity and distinct bullish, bearish, and sideways regimes through adaptive regime switching.The gating network performs conditional computation by selecting relevant experts for each market state.
  • 2) Mixture-of-Experts System:: Expert outputs are combined through a batch-wise weighted sum, enabling dynamic reconfiguration based on each batch’s latent state.The aggregation operates over expert outputs using the gating weights.
  • 2) Mixture-of-Experts System:: The gating network computes expert-selection weights from flattened inputs using a learnable MLP and Softmax operation.The MLP maps vec(X) ∈RB×LC to gating logits before Softmax produces the gating weights.
  • 3) Prediction Head:: The prediction head decodes aggregated latent features ˆX through a multi-layer perceptron to produce the final forecast.The head maps the aggregated representation into the model’s forecast output.
  • 3) Prediction Head:: The head incorporates nonlinear activations and stochastic regularization to support robust mapping and mitigate potential overfitting.These components are included to improve the robustness of the latent-to-forecast mapping.
  • 3) Prediction Head:: The prediction output Ypred ∈RB×Horizon uses GELU activation while refining high-dimensional latent representations into price or return forecasts.The structure is described as maintaining generalization capability while producing precise forecasts.
  • 2) Mixture-of-Experts System:: The overall MoFE architecture processes inputs through parallel global FNO and local Conv paths within experts before gating-based dynamic weighting.This combines global and local processing paths within the expert system.

C. Loss Function

The Break-even Optimized Loss jointly optimizes magnitude accuracy, correlation alignment, and directional consistency through differentiable objectives. An orthogonality-constrained gating regularizer promotes diverse expert activation patterns and specialization across market regimes or latent feature subspaces.

  • Break-even Optimized Loss: The Break-even Optimized Loss combines weighted MSE of log returns, Information Coefficient loss, and Directional Accuracy loss to capture market dynamics beyond Euclidean error.The hyperparameters α and β govern trade-offs among these objectives.
  • Break-even Optimized Loss: Magnitude Accuracy is weighted by α, while Directional Frequency uses (1 −α −β) for LDir and β for LIC within a differentiable formulation.This preserves continuous optimization while balancing the loss components.
  • Gating Regularization: MoE expert collapse occurs when gating repeatedly activates the same subset of experts, sacrificing model capacity.The regularization objective is introduced to enforce expert specialization and diversity.
  • Gating Regularization: The gating regularizer penalizes off-diagonal Gram-matrix correlations after L2 normalization, driving normalized activations toward identity structure.This encourages disjoint activation patterns and distinct specialization across market regimes or latent feature subspaces.

IV. EXPERIMENTAL ANALYSIS AND DISCUSSION · A. Datasets

The study uses chronologically ordered daily Bitcoin OHLCV data from January 1, 2020, to December 27, 2025, with leakage-controlled temporal splits and locally normalized windows. The MoFE feature space expands these market variables with logarithmic returns and four technical indicators, while excluding external macroeconomic inputs and acknowledging reduced reliability during black-swan events.

  • A. Datasets: The Bitcoin dataset spans January 1, 2020, to December 27, 2025, and contains daily chronologically ordered OHLCV records.Data were collected through a bespoke web crawler from historical Bitcoin repositories.
  • A. Datasets: The dataset was physically partitioned into training, validation, and testing sets at an approximate 4:1:1 ratio before preprocessing.This chronological partitioning was used to preserve temporal integrity.
  • A. Datasets: The raw Binance exchange data includes six attributes: Timestamp, Open, High, Low, Close, and Volume.The data were accessed through Binance’s v3/klines endpoint using a Python script.
  • A. Datasets: Normalization was applied locally to each sliced window, while global standardization parameters and z-scores were derived exclusively from training data.Test-time inference and metric calculations used only information available before each forecast time point.
  • A. Datasets: The methodology relies strictly on endogenous Binance OHLCV variables and excludes external macroeconomic indicators such as interest rates and inflation.The stated scope therefore omits external drivers that might influence long-term valuations.
  • A. Datasets: MoFE’s predictive reliability may diminish under extreme market conditions or systemic black-swan events.This limitation reflects the model’s dependence on historical market information and conditions represented in the data.
  • A. Datasets: The MoFE feature space is expanded to an 8-dimensional vector using logarithmic OHLC returns and RSI, MACD, RVol, and VolProxy indicators.VolProxy characterizes price fluctuations from log returns over a rolling window of n = 5 days.

B. Performance Metrics · C. Experiments Setup

The study evaluates MoFE through statistical, trading-strategy, and profitability dimensions, while enforcing consistent implementation and optimization settings across models. Experiments use controlled forecasting configurations designed to assess persistent temporal dependencies and limit overfitting.

  • B. Performance Metrics: MoFE is evaluated using statistical predictive, trading strategy, and profitability metrics.The framework is explicitly organized into three evaluation dimensions.
  • B. Performance Metrics: Statistical metrics include RMSE, MAE, and R2, while trading-strategy evaluation includes Information Coefficient (IC).These metrics assess forecast error, captured return variance, and investment-signal quality.
  • B. Performance Metrics: 0.1% transaction fees and 0.2% slippage fees are applied to day-line trading.These costs are incorporated into the experimental trading assumptions.
  • C. Experiments Setup: All models use PyTorch on an NVIDIA GeForce RTX 4060 Laptop GPU under a fixed seed of 42 and consistent data and input-feature configurations.The setup aims to ensure deterministic behavior and fair benchmarking.
  • C. Experiments Setup: AdamW is used for optimization with a standardized learning-rate schedule across all models.This provides a common optimization procedure for the evaluated methods.
  • C. Experiments Setup: ReduceLROnPlateau, weight decay, and early stopping with patience of 20 epochs are used to improve generalization, selecting the lowest-validation-loss checkpoint.These controls target overfitting and determine the model used for testing.
  • C. Experiments Setup: L = 32 days, dmodel = 48, K = 2 experts, and M = 8 frequency modes define the reported MoFE configuration, with evaluation emphasizing T +5.The longer horizon is emphasized because T + 1 predictions are highly susceptible to transient noise.

D. Price Forecasting Experiments · 1) Overall Performance: · 2) T+1 Forecasting:

MoFE is evaluated against state-of-the-art baselines at T+1 and T+5 horizons, outperforming existing methods overall. At T+1, it combines strong pointwise accuracy, directional predictive capability, noise filtering, and trading performance while mitigating phase lag.

  • 1) Overall Performance:: MoFE is benchmarked against state-of-the-art baselines across T+1 and T+5 prediction horizons.The comparative study reports results in Tables II and III.
  • 1) Overall Performance:: MoFE consistently surpasses RNN, iTransformer, CryptoMamba, FreqMoE, and MoLE baselines across the reported comparisons.LSTM and GRU achieve competitive Directional Accuracy but underperform in profitability and precision, while the other baselines lack MoFE’s comprehensive competitiveness or performance margin.
  • 2) T+1 Forecasting:: 1582.64 RMSE and 0.9827 R2 are MoFE’s minimum RMSE and maximum R2 at T+1, indicating precise mapping of continuous market dynamics.The passage contrasts this performance with conventional phase-lagged predictions approximating Pt+1 ≈ Pt.
  • 2) T+1 Forecasting:: 58.43% DA and 0.1957 IC are MoFE’s peak T+1 directional metrics, with IC exceeding the runner-up baseline by 1.6×.This indicates a robust monotonic correspondence between predictive signals and realized returns.
  • 2) T+1 Forecasting:: MoFE’s predicted trajectory closely tracks actual market trends despite dampened amplitude caused by the balanced loss function’s penalty on large directional deviations.The loss function induces strategic conservatism while preserving trend alignment.
  • 2) T+1 Forecasting:: MoFE filters high-frequency noise while preserving high-fidelity directional signals, supported by dense clustering around y = x and near y = 0.The y = x clustering indicates strong linear correlation, while the y = 0 clustering reflects the loss function’s risk-aversion effect.

3) T+5 Forecasting: · 4) Robustness and Horizon Sensitivity:

For T+5 forecasting, MoFE leads predictive accuracy, directional metrics, and financial performance, while remaining robust as the horizon extends from T+1 to T+5. Its ROI improves over the longer horizon despite general deterioration in predictive metrics across models.

  • 3) T+5 Forecasting:: MoFE records the smallest RMSE (3360.33) and highest R2 (0.9186) for T+5 price prediction, despite MAE (2593.37) being marginally higher than iTransformer.The MAE difference is reported as 1.05, while the lower RMSE indicates greater stability against large errors.
  • 3) T+5 Forecasting:: MoFE leads T+5 directional prediction with IC (0.1389) and DA (56.10%).These directional metrics are described as similarly leading the T+1 results.
  • 3) T+5 Forecasting:: MoFE achieves the best T+5 ROI (4.91) and Sharpe Ratio (1.73), although it does not attain the highest Win/Loss Ratio.Its ROI is reported to surpass the second-best approach by a factor of three, supporting its risk-adjusted performance.
  • 3) T+5 Forecasting:: MoFE’s T+5 financial results identify it as the optimal architecture for maximizing risk-adjusted returns.This conclusion follows the reported exceptional ROI and Sharpe Ratio despite the lower Win/Loss Ratio.
  • 4) Robustness and Horizon Sensitivity:: As the horizon extends from T+1 to T+5, all models show increased RMSE and MAE alongside decreased R2.The passage characterizes this as a natural deterioration in predictive metrics.
  • 4) Robustness and Horizon Sensitivity:: MoFE exhibits superior robustness across horizons, with only a marginal and controllable decline in predictive performance.Its robustness is assessed relative to the deterioration exhibited by all models when moving from T+1 to T+5.
  • 4) Robustness and Horizon Sensitivity:: MoFE’s ROI improves rather than declines at the longer horizon, indicating its ability to capture mid-term market dynamics.The passage states that MoFE therefore dominates short-term forecasting and leads at the extended horizon.

E. Efficiency Analysis · F. Interpretability Analysis · G. Ablation Study

MoFE combines a lightweight computational footprint with interpretable expert and feature reweighting, while ablations validate regularization, two-expert routing, dynamic gating, and a 3 × 1 kernel as important design choices.

  • E. Efficiency Analysis: E. Efficiency Analysis: MoFE uses 93.6K parameters and 2.36M FLOPs for T+1, versus 94.1K parameters with identical FLOPs for T+5.These efficiency metrics are described as highly competitive with established baselines.
  • E. Efficiency Analysis: E. Efficiency Analysis: MoFE achieves SOTA performance with a lightweight footprint despite its dual-domain structure, competing with more parameter-heavy iTransformer and Mamba-based models.The comparison is framed in terms of higher memory and computational overhead for those baselines.
  • F. Interpretability Analysis: F. Interpretability Analysis: Dynamic expert weights shift across conditions, assigning short-term fluctuations and long-term trends to distinct experts.Regularization loss encourages expert specialization and provides both time- and frequency-domain perspectives.
  • F. Interpretability Analysis: F. Interpretability Analysis: The eight input features receive condition-dependent weights, with stronger recalibration when Bitcoin-market volatility is high.The features have roughly equal weights in most cases before dynamic adjustment under high volatility.
  • G. Ablation Study: G. Ablation Study: Removing Lreg markedly reduces DA and Win/Loss Ratio, while marginally decreasing Sharpe Ratio and ROI and permitting expert collapse.The results support Lreg’s role in specialization and more accurate modeling of diverse market dynamics.
  • G. Ablation Study: G. Ablation Study: N=2 performs best, whereas N=1 or N=4 degrades most metrics; N=4 causes expert dilution and N=1 cannot resolve high-frequency and long-term patterns simultaneously.The 32-day look-back window provides insufficient supervision for a high-dimensional expert space, while N=2 balances specialized temporal scales.
  • G. Ablation Study: G. Ablation Study: Replacing dynamic gating with averaging significantly worsens IC, DA, and profitability metrics, confirming gating as a regime-sensitive expert decision-maker.The ablation retains Lreg while isolating the contribution of dynamic routing.
  • G. Ablation Study: G. Ablation Study: A 3 × 1 convolutional kernel is optimal, while 1 × 1 loses local temporal correlations and 5 × 1 introduces noise from distant time steps.The tested kernel sizes were 1 × 1, 3 × 1, and 5 × 1.

V. CONCLUSION

MoFE integrates Fourier Neural Operators and Mixture-of-Experts with dual-domain modeling and dynamic gating to capture cryptocurrency markets’ non-stationary, multi-scale stochasticity and regime shifts. The framework achieves SOTA T+1 and T+5 forecasting, mitigates phase lag, performs robustly during high volatility, and delivers high risk-adjusted returns in simulated trading, while future work targets cross-asset volatility modeling.

  • Core framework: MoFE integrates FNOs and MoE to model cryptocurrency markets’ non-stationary, multi-scale stochasticity.Its dual-domain architecture extracts global spectral trends and local temporal microstructures, while dynamic gating adapts to rapid regime shifts.
  • Forecasting performance: MoFE achieves SOTA results for T+1 and T+5 forecasting horizons while reducing phase lag and enhancing directional predictive power.
  • Forecasting performance: Frequency-domain modeling separates high-frequency noise from structural market trends, explaining MoFE’s phase-lag mitigation.This advantage is particularly robust during high-volatility periods, when spectral representations isolate structural trends from transient short-term noise.
  • Trading utility: Simulated trading shows that MoFE delivers exceptional risk-adjusted returns and high practical utility.
  • Future research: Future research will extend MoFE to other financial time-series forecasting tasks, including cross-asset volatility modeling.
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