Given a physical data set, one of the most important questions one can pose is: “Can we predict the future?” This question can be put forward irrespectively of the fact that we may already have some insight or even be certain on what the exact model behind some or all the observed variables is. For example, for chaotic dynamical systems [1, 2], we may even have the underlying dynamics but still find it hard to predict the future, given that chaotic systems have exponential sensitivity to initial conditions. The more chaotic a system is (as measured by the positiveness of their largest Lyapunov exponents [3, 4]) the harder it gets to predict the future, even within very short time horizons. In the limit case of a random system, it is not possible to predict the future at all, although one can opine on certain future statistics. For the case of weakly chaotic systems, there is an extensive literature on forecasting methods ranging from linear approximations; truncated functional expansion series[7, 8]; non-linear embeddings ; auto-regression methods11]
to state-of-the-art neural networks and deep learning methodologies and many others, too long to list here.
Most literature on forecasting chaotic signals is dedicated to a single time series, or treat a collection of related time series as a non-extended set, i.e. a multivariate set of discrete variables as opposed to a spatially continuous series. For forecasting spatial-temporal chaos we refer the reader to [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and references therein. Even rarer are attempts to forecast spatial-temporal chaos using neural networks and deep learning methodologies [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]
, although this field of research is clearly growing at the moment111 There is also a new emerging field of research on solving PDEs (therefore implicitly predicting a spatial-temporal evolution) using deep learning – see [35, 36, 37]
and references therein. Furthermore, notice that in this article we are concerned with the full space-time prediction, as opposed to the ongoing research on pattern recognition in moving images (2D and 3D), which attempt to pick particular features (e.g. car, pedestrian, bicycle, person, etc.) and to forecast where those features will be in subsequent images within a particular moving sequence – see and reference therein. . Nonetheless, this area of research is of importance, as most physical systems are spatially extended, e.g. the atmospheric system driving the Earth’s weather ; the solar dynamo driving the Sun’s sunspots ; and the influence of sunspots on the Earth’s magnetic field via the solar wind, coronal mass ejections and solar flares – the so-called space weather [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54], which may have real economic implications . Nonetheless its importance, forecasting spatial-temporal chaos is difficult. The reasons are many, but mainly: first, the geometric dimension of the attractor 
– usually quite large, the so-called curse of dimensionality; and second how to choose the variables to use for forecasting, i.e., is there enough information on the same point back in time to derive the future of that particular point, or do spatial correlations and spatial propagation affect it in a way that one must take into account some spatial and temporal neighbours set to forecast the future. If this is the case, can that set of points be defined and how can it be constructed? It is this last question that we investigate in this article, in the particular context of spatial-temporal forecasting using neural networks.
for examples and references). For forecasting of time series, a simple approach consists of designing the input layer as a vector of previous data using a time delay, the time delay neural network method[59, 60, 61, 62, 63, 64, 65]. For spatial-temporal series, one can generalize it to include temporal and spatial delays [22, 24]. This is where the connection to dynamical systems can be useful.
In 1981, Takens established the theoretical background  in his embedding theorem for a mathematical method to reconstruct the dynamics of the underlying attractor of a chaotic dynamical system from a time ordered sequence of data observations. Notice the reconstruction conserves the properties of the original dynamical system up to a diffeomorphism. Further developments established a series of theorems [67, 68, 69] that provided the basis for a non-linear embedding and forecasting on the original variables. The theorems and related articles propose to use a time delay approach with the time lag based on the first minima of the mutual information222 Notice that another non-linear dynamical systems technique exists to calculate this time delay, the zero of the autocorrelation function [70, 71], but essentially these two approaches are after the same objective, i.e. to select uncorrelated variables as much as possible for optimal reconstruction embedding. So, in this article, we focus only on the first minima of the mutual information for simplicity of analysis. – see [72, 71, 70] – and to choose the number of points to include using the method of false nearest neighbours detection suggested by  and reported in detail in [74, 75, 76, 71].
Some authors discuss the use of either the mutual information and/or embedding dimension as a constraint on feature representation [77, 78, 79, 80, 81, 82, 62, 83, 84, 85, 86, 87, 88, 65, 89, 64, 90, 91, 92, 93, 94, 95, 96, 97, 83, 79, 93, 78, 94]. Others  attempted to generalize the mutual information approach to higher dimensions but do not actually connect it to the problem of spatial-temporal forecasting using neural networks. There are also authors  that try to use neural networks to determine the optimal embedding and time delay for the purpose of local reconstruction of states with a view to forecast (the opposite of what we try to empirically demonstrate here). Fig. 6 in 
shows how the forecasting error for a pure time series prediction changes with the delay and the number of time delay points used as an input – they use a reinforcement learning based dimension and delay estimator to derive the best dimension and delay, but do not seem to show that it is the dynamical systems’ derived values that are indeed optimal for forecasting neither they show any extension to spatial-temporal signals as we demonstrate in this article. Other authors
try to use Support Vector Machines (SVMs) to forecast spatial-temporal signals and use delays and embedding approaches to define the state vectors. In fact, Parlitz and Merkwirth
mention that local reconstruction of states “…may also serve as a starting point for deriving local mathematical models in terms of polynomials, radial basis functions or neural networks…”. Here we attempt to show empirical evidence that this is not just a starting point, but the optimal neural network input feature selection.
We also emphasise that, as far as we are aware, all the references on neural network forecasting of spatial-temporal dynamics that use the embedding theorems and the related mutual information and the false nearest neighbours methods seem not to justify its use, i.e., the approach is explained, even suggested to be optimal, but neither proven theoretically or empirically. Here we attempt to provide an empirical evidence for this optimality. Using this theoretical framework, we propose that this non-linear embedding method, using the training data alone without reference to the forecasting model, can be used to indicate the best way to construct the feature representation for the input layer of a neural network used for forecasting both in space and time. In order to support this proposal, we, in this article, show empirical evidence for an optimal feature selection for four particular cases of two-dimensional spatial-temporal data series , where by two-dimensional we mean a scalar field that can be defined by a matrix with components . Furthermore, notice the primary goal is not to demonstrate the ability to forecast, which has already been done by several authors in the literature above, but rather that there is no need to calibrate the neural network feature selection specification by the “dark art” of trial and error.
The neural network architecture we chose to demonstrate our proposal is a form of the basic feedforward neural network, sometimes called the time-delayed neural network , trained using the so-called back-propagation algorithm [101, 102, 103, 104]. We focus on spatial-temporal series, so we have extended the usual time-delayed neural network to be a time and space delayed network. The overall feature representation of the network is depicted in detail in Fig. 1
. Notice we chose to use feedforward neural networks rather than more complex neural networks such as recurrent neural networks, since feedforward ones are simpler to design; are capable of being used for forecasting of even complex chaotic signals; are guaranteed to converge, at least, to a local minima; and are easier to interpret.
Under this input representation, we use the ideas proposed in [17, 18] to construct a grid of input values which are then fed to the neural network to produce a single output, the future state. Formally, let and . Consider a spatial-temporal data series which can be defined by a matrix with components . To these components, we will call states of the spatial-temporal series. Consider a number of neighbours in space of a given and a number of temporal past neighbours relative to (see Fig. 1 for details). For each , we define the input (feature) vector with components given by , its spatial neighbours and its past temporal neighbours, and with and being the spatial and temporal lags:
So, the input is a vector
and the target (output) to train the network is the value. We train the network using stochastic gradient back-propagation by running a stochastic batch where we randomly sample pairs of inputs and outputs from the training set: and , respectively. Then at test time we chose inputs , such that , being the number of temporal slices on the training set. As for the remaining architecture, we use one hidden layer with
nodes. Regarding the back-propagation hyperparameters, we included an adaptive learning rate, where the hyperparameter is the initial learning rate and is the learning rate used at time step . We included a momentum for faster convergence. A further hyperparameter is the choice of the activation function (see 
), we use either a ReLu (rectified linear unit) or a logistic sigmoid function depending on the test case we are working with. We also normalize the data before passing it through the neural network, in most cases we scale it in linear fashion, and in the case of real physical data as we will see later, we scale it in logarithmic fashion it by , where is the initial data, and and are the arbitrary shift and scaling constants, respectively. For the weight (and bias) initialization we chose random numbers with a constant distribution between and shifted by and scaled by . All of these hyperparameters are calibrated and fixed before we do any simulations with respect to the parameters , , , , which are auto-calibrated by the above mentioned methods derived from dynamical systems theory. In this sense, , , ,
are not hyperparameters of the neural network. We use the standard loss functionfor a prediction centred around using as input the feature vector of total dimension .
Once we do a forecast, we then compare the goodness of fit by first visual inspection and second by numerically calculating the so-called structural similarity which has been proposed by  and used already in the context of spatial-temporal forecasting in [22, 24]. It has also been used in the context of deep learning used for enhancing resolution on two dimensional images  and restoring missing data in images . For details on the SSIM measure see [107, 110, 111]. The SSIM index is a metric quantity used to calculate the perceived quality of digital images and videos. It allows two images to be compared and provides a value of their similarity - a value of corresponds to the case of two perfectly identical images. We use it by calculating the between the entire test set and the forecast set, since these can be interpreted as images (one spatial dimension/one temporal dimension).
Here we propose that the optimal time delay/spatial delays ( and , respectively) must be the ones based on the first minima of the mutual information [72, 71, 70] and that the optimal number of temporal/spatial points to use ( and , respectively) must be the ones based on the method of false nearest neighbours detection [73, 74, 75, 76, 71]. The mutual information is calculated by taking a , a one-dimensional data set, and , the related -lagged data set. Given a measurement , the amount of information is the number of bits on , on average, that can be predicted. We then average over space and take the first minimum of , or, in the absense of a clear minimum, take the temporal lag for which the drops significantly and starts to plateau. This calculates , the optimal time delay. Conversely, we calculate by calculating the spatial lag for which we obtain the first minima of the time-averaged mutual information . Once the optimal spatial and temporal lags and are calculated, we calibrate the minimum embedding dimension, or in other words, the number of spatial and temporal neighbours in optimal phase space reconstruction. We use the method of false neighbours [73, 74, 75], which determines that falsely apparent close neighbours have been eliminated by virtue of projecting the full orbit in a increasing higher dimensional embedding phase space. This gives us the , the optimal number of time slices to take, and , the optimal number of spatial slices to take in our optimal reconstruction.
In this article, we propose that as any set of input representation “approaches” the optimal one, then . In the case of finite training sets and/or noisy training sets , where
is the best forecast possible given the data set. Visually, we believe that the SSIM versus some reasonable metric constructed to represent the distance between any input representation and the optimal input representation will show a skewed bell shape as depicted in Fig.2. In this proposal, we use the most obvious candidate to represent the distance between any input representation and the optimal input representation, the Euclidian distance given by , where ,,, are the parameters for each representation and ,,, are the ones derived from the dynamical systems theory. We also verified that other reasonable metrics, in particular the Manhattan distance, did not change the results qualitatively.
In order to empirically substantiate our proposal, we take four examples of spatial-temporal series and attempt to forecast using our feedforward neural network. First, we split the data into a training and a test set. Second, using the training set only, we calculate the optimal time delay/spatial delays ( and , respectively) using the first minima of the mutual information, and then we calculate the optimal number of temporal/spatial points to use ( and , respectively) using the method of false nearest neighbours. Only then we build the neural network model, calibrating the hyperparameters of the network by exhaustive search on the parameter space to minimize the error on the training set. Then having fixed those hyperparameters, we use a Monte Carlo simulation on the test set, on each one of our four examples, sampling random values of the key feature selection parameters: , , , (including the trivial ones with and/or ) and calculate the values and . We plot the latter as a function of the former to compare against our proposal as depicted in Fig. 2.
We first take a physical system example, a real data example, and then we progress from “simpler” systems (coupled maps) capable of generating spatial-temporal chaos to more “complex” systems (coupled Ordinary Differential Equations - ODEs) to “really complex” systems (Partial Differential Equations - PDEs). This is partially motivated by results in the literature that show that general universalities are present in different levels of simplification of physical models , from the original PDEs to truncated ODE expansions (e.g. spectral method expansions ) to the most extreme simplification or discretization such as maps which capture the essence of the problem(s). In all cases we take examples with one spatial and one temporal dimension. However, we believe that our proposal will extend to multiple spatial dimensions. Again, notice that here we are not trying to demonstrate that neural networks, and in particular feedforward neural networks can perform well in predicting spatial-temporal chaos (as this has been demonstrated in the literature already), but rather to show that the optimal choice of the input layer features is given by dynamical systems theory and does not need to be another neural network hyperparameter calculated by the “dark art” of trial and error.
Iv-a Sunspot data - a physical system example
The first example we take is a physical real data example based on a previous article of one of us , where a neural network using the type of input representation above (Fig. 1) was used to forecast sunspot areas in our Sun in both space ( latitude) and time (Carrington Rotation index333Given that the surface solar rotation varies with time and latitude, any approach of comparing positions on the Sun over a period of time is necessarily subjective. Therefore, solar rotation is arbitrarily taken to be 27.2752316 days for the purpose of Carrington rotations. Each solar rotation is given a number, the so-called Carrington Rotation Number, starting from 9th November, 1853.). This sunspot data is usually called the “butterfly diagram” due to its butterfly wings like appearance . One can see how this butterfly diagram looks like in .
Sunspot data is regularly seen as a benchmark for time series forecasting, given its chaotic nature and that it considered to be among the longest continuously recorded daily measurement made in science . Many authors [118, 119, 120, 121, 122, 123, 96, 97, 124, 91, 125, 126, 83, 127, 128, 90, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 88, 153, 93, 154, 155, 156, 157, 158, 78, 159, 160, 94] have already attempted to use neural networks to forecast aspects of the sunspot cycle, although as far as we are aware, none in both space and time having restricted themselves to using these neural networks to forecast mostly either the sunspot number or the sunspot areas as a function of time. There is only one example, as far as we are aware, of actual spatial-temporal forecasts using neural networks (see also [144, 145] where a neural network forecast of the magnetic flux, which is related to sunspots, is forecast for latitude/longitude datasets). There are also a few examples of forecasting the butterfly diagram sunspot data in both space and time (latitude/time) [161, 162, 163, 164, 22, 165] but none of these used neural networks, rather all of those used other statistical methods or numerical physical modelling.
We take as a “training set” the data from the year 1874 to approximately 1997 (i.e. the first 1646 Carrington Rotations). We then attempt to reproduce or forecast the sunspot area butterfly diagram from Carrington Rotation 1921 up to 2162 (the last one corresponding approximately to the year 2015); that is, we use 1646 time slices ( years) to reproduce the next 242 time slices ( years)444We use exactly the same training set/forecast set slicing as in [22, 24] for consistency, even if more data is already available at this time.. The training set corresponds to around 12 solar cycles (cycle 11 to 22), while the “forecasting set” equates to around 1.5 cycles (cycle 23 and half of cycle 24). The entire dataset, including the training and forecasting sets, is a grid , with and . The training set is a grid . For this case the optimal values were , , and as calculated in . The hyperparameters of the neural network were: , , , a logarithmic normalization of the inputs scaled with and , weight initialization with and and . We used the logistic sigmoid function as the activation on both the hidden and output layers.
The Monte Carlo results are depicted in Fig. 3 showing runs with different , , , and plotting the SSIM versus the distance to the optimal input feature selection parameters (,,,) given by the dynamical systems theory. It shows a reasonable expected dispersion as proposed and a good convergence to the highest SSIM value we could obtain for this particular slicing of the training and forecast sets . From the figure, there seems to be also two clusters of behaviour, and at closer inspection, we found that the cluster with lower SSIM is basically a set of very bad forecasts, with none of the characteristics of the real sunspot behaviour (the 11 year-like cycle and the migration to the latitudinal equator), while the higher SSIM cluster corresponds to visually recognizable sunspot butterfly-like diagrams.
These results were quite satisfactory and inspired us to attempt to check the existence of a universality of behaviour across dynamical systems, by examining other unrelated synthetic generated data sets. We continue below to these attempts.
Iv-B Coupled Hénon maps - a discrete-time dynamical system
Motivated by having a real case from a physical system, we then tried to investigate if this same proposal holds in a very simplified example of a spatial-temporal model. Coupled maps are widely used as models of spatial-temporal chaos and pattern/structure formation [166, 167, 168]. Following [18, 17] we take a lattice of coupled Hénon maps:
with fixed boundary conditions and . The initial values for rest of the variables and is taken from a random constant distribution in the range .
We run the synthetic data generation for time steps, and divided the set into time steps for the training set and time steps for the test set. The other parameters of the neural network were: , , , a linear input normalization scaling with , , , and . We used the ReLu function as the activation on both the hidden and output layers.
For this case the optimal values given by the mutual information and the false neighbours methods were , , and . The results of the Monte Carlo simulation for different , , and are depicted in Fig. 3. It again shows a dispersion as proposed and a reasonable convergence to the highest SSIM value we could obtain for this particular slicing of the training and forecast sets .
Results suggest the same structure as depicted in our proposal diagram and in the previous results for sunspots. We now move below to a more complex model, a coupled set of ODEs.
Iv-C Coupled Ordinary Differential Equations - Lorenz-96 model
For the spatially extended coupled ODEs model we used a well-known 40-coupled ODE dynamical system proposed by Edward Lorenz in 1996 :
where , and and is a forcing term. We use the forcing to get some interesting behaviour in space and time. We used a time step and we have integrated this equation using J. Amezcua’s MATLAB code as given in . It uses the Runge-Kutta 4-step method.
We run the synthetic data generation for time steps, and divided the set into time steps for the training set and time steps for the test set. The other parameters of the neural network were: , , , a linear normalization input scaling with and , weight initialization with and and . We used the ReLU function as the activation on both the hidden and output layers.
For this case the optimal values obtained before the Monte Carlo simulation from the mutual information and false neighbours methods were , , and . The results of the random sampling of , , , in the simulation are depicted in Fig. 5. It shows a dispersion as proposed and a quite a good convergence to the highest SSIM value we could obtain for this particular slicing of the training and forecast sets: . Results suggest the same structure as depicted in our proposal diagram and in the previous results for sunspots and the coupled Hénon maps.
Iv-D Partial Differential Equations - Kuramoto-Sivashinsky model
Finally we take a full PDE system, the Kuramoto-Sivashinsky model [171, 172], a very well-known system capable of spatial-temporal chaos and complex spatial-temporal dynamics. It is a fourth-order nonlinear PDE introduced in the 1970s by Yoshiki Kuramoto and Gregory Sivashinsky to model the diffusive instabilities in a laminar flame front. The model is described by the following equation:
where with a period boundary condition . The nature of solutions depends on the system size and on the initial . We have integrated this equation by taking an exponential time difference Runge-Kutta 4th order method (ETDRK4) using the Matlab code by P. Cvitanović as given in  and taking a time step of , Fourier modes which are known to produce a “turbulent” or chaotic behaviour and a initial condition for , the remain being .
We run the simulation for time steps, and divided the set into time steps for the training set and time steps for the test set. The other parameters of the neural network were: , , , a linear normalization input scaling with and , weight initialization with and and . We used the ReLU function as the activation on both the hidden and output layers.
The results of the Monte Carlo simulation can be seen in Fig. 6. For this case the optimal values obtained before we run the Monte Carlo simulation were , , and . It again shows a dispersion as proposed and a excellent convergence to the highest SSIM value we could obtain for this particular slicing of the training and test sets: a surprising high value of . Results suggest the same structure as depicted in our proposal diagram and in the previous results for sunspots, the coupled Hénon maps and coupled ODEs.
In this paper, we have shown empirical evidence for the existence of an optimal feature selection for the input layer of feedforward neural networks used to forecast spatial-temporal series. We believe that the selection of the features of the input layer can be uniquely determined by the data itself, using two techniques from dynamical systems embedding theory: the mutual information and the false neighbours methods. The former procedure determines the temporal and spatial delays to take when selecting features, while the latter determines the number of data points in space and time to be taken as inputs. We propose that this optimal feature selection gives the best forecast, as measured by a standard image similarity index. We also propose that the shape of the dispersion of points on a Monte Carlo simulation across all possible feature selections on a plot of the similarity index versus the distance to optimal feature selection is a skewed bell shape with the highest value being the optimal feature selection/maximum similarity index.
In order to substantiate our proposal, we chose four unrelated systems, in order of complexity: a set of spatially extended coupled maps; a set of spatially extended coupled ODEs; a one-dimensional spatial PDE and a real spatial-temporal data set from sunspots areas in our Sun. In all four cases, we were able to first use the mutual information and the false neighbours methods to determine the four parameters defining the input layer feature selection555We have four parameters for the feature selection in these cases, with one temporal and one spatial dimension. For higher dimensional systems, there will be more parameters, the exact number being double the number of dimensions of the system.. After calibration of the hyperparameters we then were able to forecast reasonably the test set, although this is not the objective or primary goal of this article. We then show that for a random Monte Carlo simulation across possible feature selections, the neural network did not, as expected, forecast as well as it did for the specific set of optimal four parameters given by dynamical systems theory. As proposed, the Monte Carlo simulations show that the shape of the distribution of points was a skewed bell shape with the highest value being the optimal feature selection/maximum similarity index (subject to minor variations due to noise and the finiteness of the dataset).
Given how important spatial-temporal systems are and how we want to forecast the future as accurately as possible it is quite important to attempt to reduce the number of hyperparameters in neural network prediction, and to try to constrain the feature selection from the data properties only. If indeed our proposal turns out to be true, it would remove the input layer feature selection as another free parameter in the already complex process of choosing the details of the neural network to use for forecasting.
In this article we have focused first and foremost in establishing empirical evidence for our proposal, within a simple framework of feedforward neural networks with one hidden layer for the purpose of prediction in one spatial and one temporal dimensions. Naturally, there are many clear extensions to our research. First to use deeper networks with additional hidden layers to possibly tackle systems which are hyperchaotic (i.e. with multiple positive Lyapunov exponents). Second, to attempt to extend the proposal with empirical evidence in high dimensions, e.g. 3+1-dimensional weather systems. Third, to extend the proposal to other commonly used neural network models, such as recurrent neural networks, particularly echo state networks [174, 25]
and long short-term memory networks. Fourth and last but not least, to demonstrate the proposal rigorously would show how dynamical systems theory can clarify the so called “dark art” in neural network feature construction. These objectives are however, outside the scope of this research article and will be pursued as part of future work.
We would like to thank Prof. Reza Tavakol from Queen Mary University of London for very useful discussions on forecasting. We also thank Dr. David Hathaway from NASA’s Ames Research Centre for providing the sunspot data on which some of the results in this article are based upon. CITEUC is funded by National Funds through FCT - Foundation for Science and Technology (project: UID/Multi/00611/2013) and FEDER - European Regional Development Fund through COMPETE 2020 - Operational Programme Competitiveness and Internationalization (project: POCI-01-0145-FEDER-006922). EB is supported by a UK RAEng Research Fellowship (RF/128).
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