Auto-Configured Networks for Multi-Scale Multi-Output Time-Series Forecasting

Yumeng Zha; Shengxiang Yang; Xianpeng Wang

cs.LGApr 8, 2026

Auto-Configured Networks for Multi-Scale Multi-Output Time-Series Forecasting

Authors: Yumeng Zha, Shengxiang Yang, Xianpeng Wang

Organizations: National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, Shenyang, China · Innovative Artificial Intelligence Cluster, Digital Future Institute, De Montfort University, Leicester, U.K. · Key Laboratory of Data Analytics and Optimization for Smart Industry (Northeastern University), Ministry of Education, Shenyang, China

Abstract

Industrial forecasting often involves multi-source asynchronous signals and multi-output targets, while deployment requires explicit trade-offs between prediction error and model complexity. Current practices typically fix alignment strategies or network designs, making it difficult to systematically co-design preprocessing, architecture, and hyperparameters in budget-limited training-based evaluations. To address this issue, we propose an auto-configuration framework that outputs a deployable Pareto set of forecasting models balancing error and complexity. At the model level, a Multi-Scale Bi-Branch Convolutional Neural Network (MS--BCNN) is developed, where short- and long-kernel branches capture local fluctuations and long-term trends, respectively, for multi-output regression. At the search level, we unify alignment operators, architectural choices, and training hyperparameters into a hierarchical-conditional mixed configuration space, and apply Player-based Hybrid Multi-Objective Evolutionary Algorithm (PHMOEA) to approximate the error--complexity Pareto frontier within a limited computational budget. Experiments on hierarchical synthetic benchmarks and a real-world sintering dataset demonstrate that our framework outperforms competitive baselines under the same budget and offers flexible deployment choices.

Figures & tables

Figure 1: Evaluator-driven auto-configuration framework for multi-scale multi-output time-series forecasting. PHMOEA proposes a configuration $x$ , and the evaluator decodes $x$ to configure preprocessing, MS–BCNN, and training, returning $F(x)$ for selection and iteration. Solid arrows denote data or prediction flows, and dashed arrows denote configuration flows.

Figure 2: IGD curves along evolution on two H-DTLZ benchmark problems.

Algorithm	IGD $\downarrow$	HV $\uparrow$	FEs $\downarrow$
PHMOEA	0.0163 $\pm$ 0.0026	0.986 $\pm$ 0.005	1113.6 $\pm$ 1.9
MOEA/D	0.1400 $\pm$ 0.0150 +	0.781 $\pm$ 0.020 +	1500
NSGA-II	0.0308 $\pm$ 0.0040 +	0.984 $\pm$ 0.006 =	1500
NSGA-III	0.0282 $\pm$ 0.0035 +	0.986 $\pm$ 0.005 =	1500
SMS-EMOA	0.0175 $\pm$ 0.0030 =	0.980 $\pm$ 0.006 =	1500

Table 1: Sintering-task results of multi-objective evolutionary algorithms (mean

\pm

std).

Model	NMSE $\downarrow$	NMAE $\downarrow$	MAPE(%) $\downarrow$
MS-BCNN	$0.268\pm 0.217$	$0.348\pm 0.138$	$1.19\pm 0.04$
OB-ISSID	$0.304\pm 0.201$	$0.381\pm 0.135$	$1.31\pm 0.07$
Ventingformer	$0.279\pm 0.199$	$0.366\pm 0.136$	$1.25\pm 0.07$
Transformer	$0.456\pm 0.177$	$0.483\pm 0.103$	$1.65\pm 0.17$
LSTM	$0.380\pm 0.220$	$0.421\pm 0.142$	$1.44\pm 0.08$
GRU-PLS	$0.462\pm 0.171$	$0.512\pm 0.082$	$1.74\pm 0.08$

Table 2: Overall forecasting performance on the real-world sintering dataset with randomly shuffled samples (mean

\pm

std).

Figure 3: Error scatter plot on the real-world sintering task under the chronological evaluation setting.

Model	NMSE $\downarrow$	NMAE $\downarrow$	MAPE(%) $\downarrow$
MS-BCNN	$4.446\pm 2.193$	$1.682\pm 0.490$	$3.02\pm 0.66$
OB-ISSID	$119.7\pm 101.7$	$8.003\pm 3.659$	$13.57\pm 7.50$
Ventingformer	$4.789\pm 4.111$	$1.698\pm 0.746$	$2.89\pm 0.87$
Transformer	$7.581\pm 8.678$	$2.180\pm 1.304$	$3.43\pm 0.76$
LSTM	$114.8\pm 206.4$	$6.334\pm 7.545$	$4.23\pm 1.53$
GRU-PLS	$10.714\pm 6.280$	$2.795\pm 1.053$	$4.61\pm 0.18$

Table 3: Overall forecasting performance on the real-world sintering dataset under the chronological evaluation setting (mean

\pm

std).

Panel of Figure 4 — Figure 4: Ablation results of the proposed methods. Panel (a) reports IGD for PHMOEA, averaged over two synthetic benchmarks (H-DTLZ2 and H-DTLZ7), while panel (b) reports NMSE on the real-world sintering dataset for MS–BCNN. For both metrics, lower values indicate better performance.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Variable	Name	Type	Range / Candidate Set
$x_{1}$	Resampling operator $\mathcal{R}$	Discrete	{ linear , decimate_repeat , hybrid , pool , conv_blurpool , fir_lowpass }
$x_{2}$	Pooling type (active if $x_{1}=\texttt{pool}$ )	Conditional Discrete	{ avg , max , median , weighted }
$x_{3}$	Aligned length $L_{p}$	Discrete	{8, 12, 24, 36, 48}
$x_{4}$	Batch size	Discrete	{16, 32, 64, 128}
$x_{5}$	Normalization layer	Discrete	{ BatchNorm , LayerNorm , InstanceNorm }
$x_{6}$	Projection channels $C_{0}$	Discrete	{8, 16, 32, 64}

Appendix

Table 4: Decision variables in PHMOEA.

Category	Variables / Targets
Raw-material composition and blending	Mixed-material moisture, temperature, and granularity; fuel granularity; coarse fraction ratio (particle size $>5$ mm); ore-type proportions in the blend (six ore types); second-mixing additions (e.g., quicklime, return fines, coke breeze); planned blending ratios; planned basicity; raw-material price indicators (five series).
Process and thermal	Temperatures (ignition, exhaust gas, BTP); pressures (furnace negative pressure, main-line negative pressure, gas pressure, air pressure); flows (coal-gas flow, air flow); ore-bin scale flow rates (six bins).
Equipment state and environment	Material bed thickness; plate-belt speed; sintering machine speed; finished-product weighing; production output.
Output targets	TFe; FeO; SiO 2 ; CaO; Basicity.

Appendix

Table 5: Input variables and output targets in the sintering dataset (grouped by category).

Benchmark	Method	IGD $\downarrow$	HV $\uparrow$
H-DTLZ2	PHMOEA	0.0051725 $\pm$ 0.00008	0.4188035 $\pm$ 0.00024
	MOEA/D	0.0043681 $\pm$ 0.00011	0.4192696 $\pm$ 0.00041
	NSGA-II	0.0049416 $\pm$ 0.00038	0.4194225 $\pm$ 0.00062
	NSGA-III	0.0041721 $\pm$ 0.00002	0.4197837 $\pm$ 0.00015
	SMSEMOA	0.0051574 $\pm$ 0.00012	0.4208402 $\pm$ 0.00004
H-DTLZ7	PHMOEA	0.0033302 $\pm$ 0.00013	0.5566254 $\pm$ 0.00001

Appendix

Table 6: Per-benchmark evolutionary results on H-DTLZ2/7 at the final generation (

g{=}100

). Best IGD (lower) and best HV (higher) per benchmark are highlighted in gray. Values are reported with 7 decimal places.

Figure 5: Comparison of Pareto fronts achieved by different evolutionary algorithms on the real-world sintering task in the MSE–parameter-count objective space.

Model	Target	MSE / NMSE	MAE / NMAE	MAPE (%)
MS-BCNN	TFe	$0.066887\pm 0.004481$	$0.199272\pm 0.006370$	$0.358438\pm 0.011667$
	FeO	$0.130500\pm 0.005977$	$0.248728\pm 0.007786$	$2.762939\pm 0.082877$
	SiO 2	$0.004147\pm 0.000860$	$0.047211\pm 0.003289$	$0.834882\pm 0.055914$
	CaO	$0.034386\pm 0.002732$	$0.137910\pm 0.004608$	$1.140071\pm 0.037974$
	Basicity	$0.000561\pm 0.000020$	$0.017901\pm 0.000419$	$0.840955\pm 0.019404$
	Overall	$0.267880\pm 0.217387$	$0.347816\pm 0.138165$	$1.187457\pm 0.041567$

Appendix

Table 7: Per-target forecasting performance on five sintering targets under the shuffled (approximately i.i.d.) evaluation setting (mean

\pm

std). For each target and metric, the best result is highlighted in gray.

Panel of Figure 6 — Figure 6: Qualitative comparison of ground-truth and predicted trajectories for five quality targets under the shuffled (approximately i.i.d.) evaluation setting. Each panel shows the test-set predictions of one model (TFe, FeO, SiO 2 , CaO, and Basicity).

Model	Target	MSE / NMSE	MAE / NMAE	MAPE (%)
MS-BCNN	TFe	$0.387834\pm 0.141648$	$0.529125\pm 0.106408$	$0.939281\pm 0.188755$
	FeO	$0.205710\pm 0.046634$	$0.344918\pm 0.047771$	$3.767097\pm 0.507709$
	SiO 2	$0.092268\pm 0.037270$	$0.256975\pm 0.064096$	$4.688675\pm 1.153858$
	CaO	$0.340601\pm 0.136297$	$0.479478\pm 0.110109$	$4.048819\pm 0.919742$
	Basicity	$0.001920\pm 0.001217$	$0.035376\pm 0.011207$	$1.639518\pm 0.517844$
	Overall	$4.445824\pm 2.193273$	$1.682359\pm 0.490118$	$3.016678\pm 0.657582$

Appendix

Table 8: Per-target forecasting performance on five sintering targets under the chronological (non-shuffled) evaluation setting (mean

\pm

std). For each target and metric, the best result is highlighted in gray.

Panel of Figure 7 — Figure 7: Qualitative comparison of ground-truth and predicted trajectories for five quality targets under the chronological (non-shuffled) evaluation setting. Each panel shows the test-set predictions of one model (TFe, FeO, SiO 2 , CaO, and Basicity).

Benchmark	Variant	IGD $\downarrow$	HV $\uparrow$
H-DTLZ2	Full (PHMOEA)	0.0051723 $\pm$ 0.00008	0.4188054 $\pm$ 0.00024
	Without De-duplication	0.0054982 $\pm$ 0.00017	0.41825782 $\pm$ 0.00029
	Without Elitism	0.0088575 $\pm$ 0.00190	0.4135749 $\pm$ 0.00629
H-DTLZ7	Full (PHMOEA)	0.0033303 $\pm$ 0.00013	0.5566297 $\pm$ 0.00001
	Without De-duplication	0.0033351 $\pm$ 0.00016	0.5466567 $\pm$ 0.00002
	Without Elitism	0.0102604 $\pm$ 0.00013	0.54902670 $\pm$ 0.00001

Appendix

Table 9: PHMOEA ablation on H-DTLZ2 and H-DTLZ7 (mean

\pm

std across random seeds).

Figure 8: IGD trajectories over generations for PHMOEA ablations on synthetic benchmarks.

Variant	Target	MSE / NMSE $\downarrow$	MAE / NMAE $\downarrow$	MAPE(%) $\downarrow$
Full (MS-BCNN)	TFe	$0.387834\pm 0.141648$	$0.529125\pm 0.106408$	$0.939281\pm 0.188755$
	FeO	$0.205710\pm 0.046634$	$0.344918\pm 0.047771$	$3.767097\pm 0.507709$
	SiO 2	$0.092268\pm 0.037270$	$0.256975\pm 0.064096$	$4.688675\pm 1.153858$
	CaO	$0.340601\pm 0.136297$	$0.479478\pm 0.110109$	$4.048819\pm 0.919742$
	Basicity	$0.001920\pm 0.001217$	$0.035376\pm 0.011207$	$1.639518\pm 0.517844$
	Overall	$4.445824\pm 2.193273$	$1.682359\pm 0.490118$	$3.016678\pm 0.657582$

Appendix

Table 10: Per-target ablation results of MS–BCNN on the real-world sintering task under the chronological (non-shuffled) evaluation setting (mean

\pm

std across 5 random seeds).

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Learning to Forecast on Multiple Spatiotemporal Hierarchies

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