On the Limits of Univariate Deep Learning for Significant Wave Height Forecasting

Yilin Zhai; Hongyuan Shi; Zaijin You

physics.ao-phSep 25, 2026

On the Limits of Univariate Deep Learning for Significant Wave Height Forecasting

Authors: Yilin Zhai, Hongyuan Shi, Zaijin You

Organizations: Shandong Key Laboratory Of Estuary And Coast & Nuclear Environment, Ludong University, Shandong, 264025, Yantai, China · School of Hydraulic and Civil Engineering, Ludong University, Shandong, 264025, Yantai, China · Institute of Coastal Research, Ludong University, Yantai, 264025, Shandong, China · Navigation College, Dalian Maritime University, Dalian, 116026, Liaoning, China

Abstract

This study conducts a systematic hyperparameter search across five deep learning architectures, DLinear, LSTM, PatchTST, ResAttLstm, and Mamba2, and nine context lengths (1-168 h) for single-station significant wave height (Hs) forecasting on NDBC buoy 41009, followed by re-evaluation of the best configurations on a 47-buoy, 37-year corpus. The five families converge to a common performance level on the multi-buoy evaluation (between-family SD = 0.0014 m^2, 0.8% of the grand mean), a spread dwarfed by the 4.83x cross-dataset MSE shift between buoy corpora. All multi-buoy trials beat persistence (mean skill +0.062), but no architecture consistently outperforms the others. On the single-buoy experiment, skill peaks at 12-24 h where five trials fall below persistence, per-family Q4/Q3 test MSE ratios range from 2.4 to 2.6, and deep models underperform persistence for the most extreme 1% of waves. These findings are consistent with the interpretation that persistence already captures the dominant linear-inertial signal in univariate Hs, and that architecture engineering under this univariate input setting has reached diminishing returns: cross-buoy variance, not model class, dominates forecast error. Future work should prioritise atmospheric covariates, zero-shot cross-buoy transfer, and decomposition of Hs into swell and wind-sea components. By establishing a rigorous reference baseline for what univariate Hs models can and cannot achieve, this study provides a benchmark against which future multivariate and physics-informed approaches can be calibrated, and offers practical guidance for lightweight buoy-level forecasting in mid-latitude storm-dominated and swell-mixed environments.

Figures & tables

Figure 1: Overview of the two-phase experimental design. This figure is a conceptual schematic generated with GPT Image 2.0 and contains no scientific data.

Figure 2: Geographic distribution of the 47 NDBC buoys used in the multi-buoy evaluation.

Property	mean $\pm$ SD	median	IQR	Q1–Q3	p95	p99	p99.9	max
41009	$1.18\pm 0.67$ m	1.01 m	0.79 m	$0.70$ – $1.49$ m	2.45 m	3.48 m	5.19 m	9.79 m
47 buoys	$1.46\pm 1.01$ m	1.21 m	1.08 m	$0.78$ – $1.86$ m	3.44 m	5.10 m	7.35 m	16.91 m

Table 1: Dataset summary. All SWH statistics are computed directly from raw NDBC hourly records.

Figure 3: Schematic of the sliding-window construction. This figure is a conceptual schematic generated with GPT Image 2.0 and contains no scientific data.

Family	Params	Key hyperparameters
DLinear	2	kernel_size [9, 93]; lr [1e-3, 9.9e-3]
LSTM	5	n_layers {2,3}; hidden {64–256}; bidirectional per layer; dropout [0.003, 0.49]; lr [1e-4, 1e-2]
PatchTST	8	patch_len {8–24}; stride {2–12}; d_model {64–256}; n_layers {1–6}; n_heads {8,16}; d_ff {128–512}; dropout [0, 0.2]; lr [1e-5, 1e-2] log
ResAttLstm	7	n_cnn {1–5}; cnn_hidden {64–384}; kernel {3–7}; attn_heads {4,8,16}; lstm_hidden {64–256}; bidirectional; lr [1e-5, 1e-2] log
Mamba2	6	n_blocks {1–4}; d_model {64–256}; d_state {16–128}; d_conv {2–4}; expand {1,2,4}; lr [1e-5, 1e-2] log

Table 2: Continuous parameters are sampled uniformly (or log-uniform for learning rate). Categorical parameters are enumerated. Batch size was fixed at 4096 for all families.

Family	Test MSE	Persistence skill	p99 MSE	Q4/Q3 ratio
PatchTST	0.0333	+0.043	0.446	2.4
DLinear	0.0335	+0.038	0.459	2.4
LSTM	0.0338	+0.029	0.516	2.5
Mamba2	0.0339	+0.026 ( $-0.015$ – $+0.048$ )	0.521	2.5
ResAttLstm	0.0343	+0.014 ( $-0.026$ – $+0.057$ )	0.558	2.6

Table 3: Per-family results on station 41009. Mean values across nine context lengths. The overall skill range (min–max across CLs) is shown for the two families with negative-skill trials.

Figure 4: Test MSE distribution across the nine context lengths per family, station 41009. Boxes show median and IQR; whiskers extend to $1.5\times$ IQR. Dashed line: grand mean. Shaded band: $\pm 1$ between-family SD.

Panel of Figure 5 — Figure 5: Persistence skill across context lengths, station 41009

Figure 6: Test MSE distribution across the nine context lengths per family, 47-buoy dataset. Boxes show median and IQR; whiskers extend to $1.5\times\text{IQR}$ .

Panel of Figure 7 — Figure 7: Persistence skill and tail performance on the 47-buoy dataset

Figure 8: Mean seasonal test MSE by family, 47-buoy dataset. Numbers above Q4 bars: per-family Q4/Q3 MSE ratio.

Figure 9: Test MSE versus persistence skill for each (family, context length) cell, 47-buoy dataset. Annotated numbers indicate context length in hours. Circled points mark Pareto-optimal configurations.

Family	Q1	Q2	Q3	Q4	$\Delta$ (Q4 $-$ Q3)
DLinear	+0.058	+0.072	+0.081	+0.064	$-0.017$
LSTM	+0.047	+0.091	+0.090	+0.050	$-0.040$
PatchTST	+0.064	+0.084	+0.087	+0.070	$-0.017$
ResAttLstm	+0.040	+0.083	+0.084	+0.044	$-0.040$
Mamba2	+0.046	+0.085	+0.078	+0.048	$-0.030$
Grand mean	+0.051	+0.083	+0.084	+0.055	$-0.029$

Table 4: Seasonal persistence skill by family, multi-buoy evaluation. Values are 9-CL means. All five families exhibit Q4 skill

<

Q3 skill.

Figure 10: Ratio of multi-buoy to single-buoy test MSE per family and context length. The dashed line marks the grand mean (4.83).

Figure 11: Persistence skill per family and context length. (a) Single-buoy. (b) Multi-buoy.

Family	MSE (0%)	MSE (5%)	MSE (15%)	Skill (0%)	Skill (5%)	Skill (15%)
PatchTST	0.1611	0.1601	0.1565	+0.0727	+0.0730	+0.0737
DLinear	0.1623	0.1613	0.1575	+0.0660	+0.0663	+0.0676
LSTM	0.1633	0.1624	0.1591	+0.0605	+0.0600	+0.0586
Mamba2	0.1640	0.1631	0.1597	+0.0567	+0.0562	+0.0551
ResAttLstm	0.1645	0.1636	0.1603	+0.0538	+0.0532	+0.0516

Table 5: Per-family 9-CL mean test MSE and persistence skill under 0%, 5%, and 15% randomly missing observations with linear interpolation, multi-buoy evaluation.

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Sampling sea state using a diffusion model

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Abstract

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