Seasonal and Quantum-inspired Models for Neutron Monitor Time Series Forecasting

1 Introduction

• LMKS records combine diurnal and seasonal cycles, long-range persistence, and transient excursions associated with solar and geomagnetic activity.These properties make reliable forecasting a challenge for scientific monitoring systems.Source: §1 · p. 1
• Short- to medium-horizon forecasts can support radiation warnings, instrument scheduling, maintenance, and downstream decision support.Forecasts and calibrated predictive intervals can also help distinguish predictable quasi-periodic behavior from unusual transients.Source: §1 · pp. 1–2
• The evaluation spans seasonal baselines, LSTM, TCN, N-BEATS, KAN, and quantum-inspired QiLSTM and QiKAN models.The comparison is intended to examine inductive biases under limited tuning budgets and typical data-quality conditions.Source: §1 · p. 2
• Seasonality-aware baselines and functional decomposition can match or outperform more complex sequence models on highly periodic monitoring series.The study presents a reproducible multi-horizon LMKS benchmark and compares these approaches under a common training regime.Source: §1 · p. 2
• “Quantum-inspired” refers to classical architectures using algebraic motifs such as complex-valued states, phase-modulated recombination, and approximate unitary constraints.The models were not executed on quantum hardware.Source: §1 · p. 2

2 Related work

• In time-series forecasting, recurrent networks such as LSTM and GRU remain fundamental for modelling temporal dependencies and gated memory, while temporal convolutional networks exploit dilated causal convolutions.Source: §2 · p. 2
• N-BEATS uses stacked fully connected blocks with backcast and forecast projections as an interpretable basis-expansion approach to multi-horizon forecasting [6].Source: §2 · p. 3
• KANs combine learned linear projections with univariate nonlinear subnetworks, operationalizing function decomposition that can be parameter-efficient for decomposable mappings [7].Quantum-inspired variants such as QKAN [8] and QLSTM [9] explore complex states, unitary-like transforms, and phase coupling, with potential optimization and hyperparameter sensitivity.Source: §2 · p. 3
• Heteroscedastic Gaussian outputs and deep ensembles address predictive uncertainty, while conformal prediction offers distribution-free recalibration with finite-sample guarantees [13] [22] [21].CRPS, negative log-likelihood, MAE, and RMSE are standard evaluation measures [15].Source: §2 · p. 3
• Li and Adelmann’s review situates forecasting for accelerator diagnostics and control around interpretability, latency, and robustness trade-offs (2022) [18].It also highlights short-latency forecasting, calibrated uncertainty for alarm thresholds, and robustness to instrumentation anomalies.Source: §2 · p. 3

3 Dataset: LMKS (Lomnický Štít Neutron Monitor)

• LMKS hourly counts show diurnal and seasonal cycles, long-range autocorrelation, and occasional solar- and geomagnetic-related disturbances.The combination supports benchmarking forecasts that must capture periodicity while remaining sensitive to rare events.Source: §3 · pp. 3–4
• The data audit removed non-numeric metadata, filled missing hourly counts forward and backward, and standardized series segments for model comparison.Irrecoverable NaN windows were dropped; training-split means and standard deviations were used for z-score normalization.Source: §3 · p. 4
• Each sliding-window example pairs T past values with the next H targets; the quick-run uses 20 hours of history to forecast 10 hours.The setting assesses both short-term responsiveness and use of longer-term seasonal structure.Source: §3 · p. 4
• Figure 1 presents the hourly LMKS series used in the experiments.Source: Figure 1 · p. 4

4 Models and implementation details

• Forecast quality is measured with MAE and RMSE under a shared PyTorch regime using Adam, early stopping, gradient clipping, dropout, and training-set z-score normalization.The initial learning rate is 10^-3, batch size is 64, and early stopping uses validation MSE with a patience of eight epochs.Source: §4 · pp. 4–5

4.1 Seasonal Naive

• Seasonal Naive copies the observation from the same hour in the previous daily cycle, providing a parameter-free, reproducible benchmark for residual structure beyond seasonality.Its seasonal period for LMKS is 24 hours.Source: §4.1 · p. 5

4.2 Long-Short Term Memory (LSTM)

• The encoder–decoder LSTM compresses a history window into a latent state and produces a multi-step forecast, with autoregressive or scheduled-teacher-forcing decoding.Deterministic forecasts use a linear readout trained with MSE; a probabilistic extension predicts Gaussian mean and positive scale parameters using Gaussian negative log-likelihood.Source: §4.2 · pp. 5–6

4.3 Temporal Convolutional Network (TCN)

• A temporal convolutional network (TCN) is a convolutional, non-recurrent sequence-modeling approach that retains causal structure, so outputs at time t depend only on past inputs, while dilated convolutions allow very large receptive fields [4].Source: §4.3 · pp. 6–7
• Residual blocks combine causal convolutions, nonlinearities, dropout, normalization, and skip connections, then map the final representation directly to the full forecast horizon.The one-shot horizon mapping avoids autoregressive decoding and supports efficient multi-step forecasting.Source: §4.3 · p. 7
• The TCN is suited to multi-step forecasting when dependencies become temporally local after dilation and autoregressive decoding is undesirable for latency reasons.Source: §4.3 · p. 7

4.4 N-BEATS

• Each N-BEATS block produces a backcast approximation and a forecast contribution; successive backcasts are subtracted from the residual, and forecasts are additively aggregated.Source: §4.4 · p. 7
• Generic fully connected blocks learn implicit basis functions, and iterative subtraction can encourage specialization into trend, seasonality, or local corrections.Block-level contributions are interpretable, while block width, depth, and count control basis-expansion richness.Source: §4.4 · p. 7

4.5 Kolmogorov–Arnold Networks (KAN)

• KAN represents a continuous multivariate function as a finite sum of univariate nonlinear functions applied to linear combinations of inputs.Source: §4.5 · p. 8
• For forecasting, KAN projects the history into scalar components, processes them with independent univariate subnetworks, and recombines their outputs into the multi-step forecast.This low-rank inductive bias can be parameter-efficient for decomposable mappings and effective on quasi-periodic signals; component count and capacity balance expressivity against overfitting.Source: §4.5 · p. 8

4.6 QiLSTM

• QiLSTM uses complex-valued hidden states and approximately unitary recurrent operators to encode phase information and rotations.Its inputs may also pass through a structured front end that produces compact, phase-aware embeddings.Source: §4.6 · pp. 8–9
• QiLSTM training can use unitarity, amplitude, or phase-drift penalties to stabilize representations of sustained oscillations and phase-locked dynamics.Exact unitary parameterizations are an alternative when exact norm preservation is required.Source: §4.6 · pp. 8–9

4.7 QiKAN

• QiKAN applies per-coordinate univariate subnetworks, combines their encodings into learned aggregator channels, and sums nonlinear contributions into a prediction.The combination coefficients may be complex, allowing constructive or destructive interference across coordinates.Source: §4.7 · p. 9
• QiKAN’s low-rank recombination can capture global nonlinear effects with fewer aggregators when cross-coordinate interactions are structured or effectively low-dimensional.Its complex variant can represent oscillatory coupling through relative phases, with structural regularizers used to support numerical stability.Source: §4.7 · p. 9

4.8 Calibration and evaluation metrics

• Point-forecast performance is assessed with Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE).The passages do not provide the full RMSE formula.Source: §4.8 · p. 9
• The evaluation covers N forecast instances, each with a horizon of length H, comparing predictions with ground truth at each horizon step.The metric notation uses predictions ŷ_i,t and ground truth y_i,t.Source: §4.8 · p. 10
• Errors are computed element-wise across horizon steps and then averaged over the full evaluation set, matching the codebase implementation.This describes the implementation’s aggregation procedure.Source: §4.8 · p. 10

Analysis

• QiKAN achieved the lowest aggregate point-forecast error in the quick-run LMKS evaluation among tested configurations.Table 1 summarizes aggregate MAE and RMSE across the forecast horizon; Figure 3 presents the summary metrics.Source: Table 1 · p. 10, §5 · p. 10, Figure 3 · p. 11
• Strong periodic priors and low-dimensional functional decompositions are effective on periodic, slowly varying LMKS data; TCN tuning and QiLSTM optimization remain sensitive.LSTM, N-BEATS, and KAN produced similar error magnitudes; the TCN result was under default quick-run hyperparameters.Source: Analysis · p. 11
• The modest-search quick run and single LMKS dataset snapshot make the results non-exhaustive; other monitor series or sampling resolutions may need further tuning and validation.The reported outcomes are a reproducible snapshot under stated defaults, not exhaustive performance bounds.Source: Analysis · p. 11

• Seasonal-naïve forecasts and other seasonality-aware methods remain competitive because they exploit dominant periodicity directly and stay stable over medium-length horizons.They also require minimal hyperparameter tuning or large training sets, making them low-cost baselines for monitoring tasks with clear periodicity.Source: §6 · p. 12
• TCNs and Transformers required careful configuration to consistently beat simpler baselines in LMKS experiments with strong seasonality and moderate horizons.Source: §6 · p. 12
• Quantum-inspired architectures show promise but are not plug-and-play; optimisation, inductive biases, regularisation, and training choices influence whether they surpass classical baselines.Source: §6 · p. 12
• On LMKS, seasonality-aware baselines and functional decomposition yield dependable results with modest complexity, while more complex models require careful tuning.Source: §6 · p. 12
• Future work will test VQC-based hybrid architectures, from high-fidelity simulations to hardware runs, using staged training, parameter-efficient ansätze, and architecture-specific regularisation.Planned benchmarks and ablations will assess whether these hybrids offer consistent expressivity or robustness benefits for scientific monitoring.Source: §6 · p. 12