Table 1.

Test Case 3: metrics of methods comparison

	NRMSE		Number of trainable parameters				Wall time (s)
	training	testing	${\mathcal{NN}}_{\mathrm{enc}}$	${\mathcal{NN}}_{\mathrm{dec}},{\mathcal{NN}}_{\mathrm{rec}}$	${\mathcal{RNN}}_{\mathrm{dyn}}$	total	offline	online
FOM								37.321
POD-DEIM (ds = 12)	4.05 ⋅ 10−1	3.92 ⋅ 10−1					797	5.839
POD-DEIM (ds = 24)	3.59 ⋅ 10−1	3.47 ⋅ 10−1					799	7.720
POD-DEIM (ds = 36)	1.71 ⋅ 10−1	1.62 ⋅ 10−1					861	7.442
POD-DEIM (ds = 48)	7.48 ⋅ 10−2	7.57 ⋅ 10−2					1124	7.976
POD-DEIM (ds = 60)	2.97 ⋅ 10−2	2.90 ⋅ 10−2					1242	8.408
AE/LSTM	1.90 ⋅ 10−1	1.98 ⋅ 10−1	8562	8651	720	17,933	11,009	0.005
AE/LSTM-e2e	2.05 ⋅ 10−2	5.87 ⋅ 10−2	8562	8651	720	17,933	33,851	0.005
AE/ODE	2.09 ⋅ 10−2	4.58 ⋅ 10−2	8562	8651	5484	22,697	23,982	0.017
AE/ODE-e2e	1.78 ⋅ 10−2	3.37 ⋅ 10−2	8562	8651	5484	22,697	97,821	0.017
LDNet	7.09 ⋅ 10−3	7.37 ⋅ 10−3	0	1480	228	1708	22,887	0.014

Training and test errors obtained with the different methods, number of trainable parameters, and wall time associated with the offline phase and online phase. Computational times are obtained on a Intel Xeon Processor E5-2640 2.4 GHz. The offline phase refers to the construction of the model: for POD/DEIM, this involves building the basis for the solution manifold and for DEIM, while for the other methods it is associated with the NN training. The online phase, instead, involves predicting the evolution of the system for a new sample once the model has been constructed. This timeframe is referred to a single sample and excludes the evaluation of the output field, given its dependence on the number of considered time and space points. Further details are provided in the main text.