## 3 Methods
### End-To-End NN-Learning
#### 3.3.1 Architecture  
Typically, two NN-based energy parametrizations are considered:  
1. First, classic convolutional auto-encoder (ConvAE) representations \(\psi(\cdot)=\phi_{D}(\phi_{E}(\cdot))\) where the encoding operator \(\phi_{E}\) maps the anomaly state \(d\mathbf{x}\) onto a lower-dimensional space and the decoder \(\phi_{D}\) has to project this encoded representation in the original space. It involves the following encoder architecture: five consecutive blocks with a Conv2D layer, a ReLu layer and a 2 \(\times\) 2 average pooling layer--the first one with 40 filters and the following four ones with two times the number of filters of the previous Conv2D layer (i.e., 80, 160 and 320 filters); and a final linear convolutional layer with 20 filters. The output of the encoder is 5 \(\times\) 5 \(\times\) 40. The decoder involves a Conv2 DTranspose layer with ReLu activation for an initial 20 \(\times\) 20 upsampling stage a Conv2 DTranspose layer with ReLu activation for an additional 2 \(\times\) 2 upsampling stage, a Conv2D layer with 40 filters and a last Conv2D layer with 22 filters (the length of the image time series times the number of covariates--the OI used in the model). All Conv2D layers use 3 \(\times\) 3 kernels. Overall, this model involves \(\approx\) 600,000 parameters.
2. GE-NN: Second, NN-based Gibbs energy (GENN) representations where \(d\mathbf{x}_{s}\), the anomaly observed at location \(\mathbf{s}\in\mathcal{D}\), is supposed to be explained by the potential function \(\psi(d\mathbf{x}_{s})\) with \(\delta\)s a predefined neighborhood of site \(\mathbf{s}\), thereby relating this representation to Markovian priors embedded in CNNs. A low energy-state \(U_{\psi}(d\mathbf{x})=\int_{\mathcal{D}}U_{\psi}(d\mathbf{x}_{s})ds\) over the entire domain \(\mathcal{D}\) ensures providing a good state space reconstruction. Regarding the architecture involved,the following scheme is used: an initial 4 \(\times\) 4 average pooling; a Conv2D layer with 40 filters, 11 \(\times\) 11 kernel, ReLu activation and a zero-weight constraint on the center of the convolution window; a 1 \(\times\) 1 Conv2D layer with 40 filters; a ResNet composed of an initial mapping to an initial 200 \(\times\) 200 \(\times\) (5 \(\times\) 40) space with a Conv2D+ReLu layer; and a linear 1 \(\times\) 1 Conv2D+ReLu layer with 40 filters. Last, a final 4 \(\times\) 4 Conv2 DTranspose layer with a linear activation for an upsampling to the input shape is considered. GE-NN involves 10 residual units for a total of \(\approx\)450,000 parameters.  
We shall point out that the considered GENN architecture is not applied to the initial 0.05\({}^{\circ}\) resolution but to grids downscaled by a factor of 4 through the introduced average pooling. First, this makes the comparison with the 0.25\({}^{\circ}\) DUACS OI resolution easier. Second, the application of GENNs to the finest resolution showed a lower performance, thereby implying that considering a scale-selection problem when applying a given prior is mandatory. The upscaling involves the combination of a Conv2 DTranspose layer with 11 filters, a Conv2D layer with a ReLu activation with 22 filters and a linear Conv2D layer with 11 filters.