A deep neural network for valence-to-core X-ray emission spectroscopy

ABSTRACT In this Article, we extend our XANESNET deep neural network (DNN) to predict the lineshape of first-row transition metal K-edge valence-to-core X-ray emission (VtC-XES) spectra. We demonstrate that – despite the strong sensitivity of VtC-XES to the electronic structure of the system under study – the DNN can reproduce the main spectral features from only the local coordination geometry of the transition metal complexes when encoded as a feature vector of weighted atom-centred symmetry functions (wACSF). We subsequently implement and evaluate three methods for assessing uncertainty in the predictions made by the VtC-DNN: deep ensembles, Monte-Carlo dropout, and bootstrap resampling. We show that bootstrap resampling provides the best performance when evaluated on ‘held-out’ testing data, and also demonstrates a strong correlation between the uncertainty it predicts and the error occurring between the target and predicted VtC-XES spectra. Finally, we demonstrate practical performance by application to unseen transition metal complexes across the entire first-row (Ti–Zn). GRAPHICAL ABSTRACT


Introduction
X-ray absorption spectroscopy (XAS) in the extended X-ray absorption fine structure (EXAFS) and the Xray absorption near-edge structure (XANES) spectral domains has been widely used across the natural sciences to provide insight into the immediate (typically < 6 Å) geometric structure around the absorbing atom and, through the information encoded in the pre-edge region of the XAS spectrum, the unoccupied valence electronic structure [1,2]. However, although XAS contains a large quantity of valuable structural information such as firstcoordination-shell bond lengths and coordination numbers, it cannot directly deliver insight into the nature of bonding, often referred to as covalency [3], neither does it have sufficient sensitivity to distinguish between ligands binding through atoms with similar atomic numbers, e.g. reignited following technical and methodological developments in tabletop laboratory X-ray generators and high-intensity synchrotron radiation, as well as advances in the fabrication of high-quality optics [7]. Importantly, for NR-XES, the X-rays interacting with the sample under study do not need to be monochromatic and precisely tunable (c.f. XAS), making the combination of NR-XES with direct structural determination techniques such as X-ray diffraction an attractive proposition [8][9][10].
K-edge emission from transition metals, i.e. the fluorescence following the refilling of a core hole in the 1s orbital of the transition metal, is the focus of the present Article and a process that yields an XES spectrum characterised by a number of distinct resonances. The strongest emission lines are the Kα 1 and Kα 2 which correspond to dipole-allowed transitions from the 2p 3/2 and 2p 1/2 levels, respectively. At higher emission energies, the 3p → 1s transitions are found; these form the Kβ 1,3 lines, and are nearly an order of magnitude weaker than the Kα transitions. The Kα and Kβ 1,3 lines collectively form the so-called 'core-to-core' XES (CtC-XES) spectra which are commonly used to retrieve the fluorescence yield XAS spectrum, and which also allows one to uncover information on ligand covalency, spin, and charge state for transition metal sites in molecules and materials [11][12][13][14]. At yet higher emission energies, the Kβ 2,5 lines are found; these form the so-called 'valence-to-core' XES (VtC-XES) spectrum and are associated with the relaxation of valence electrons to fill the core hole. While these transitions are much weaker than the Kα and Kβ 1,3 lines due to the reduced overlap between the initial and final wavefunctions, the VtC-XES spectrum displays strong sensitivity to the chemical environment of the absorbing atom. It is this strong sensitivity, coupled with the increased availability of these measurements today, that has been the driving force behind the wide uptake of VtC-XES, evidenced through a multitude of applications to molecules and materials at the first-row transition metal K-edges [15][16][17][18][19][20][21][22][23][24][25][26].
The progress in experimental XES has driven commensurate progress in theory aligned with the interpretation of the XES spectral observables [27][28][29][30]. Examples from the field of VtC-XES include work exploiting a quasi-one-electron approach [15], linear-response time-dependent density functional theory (LR-TDDFT) in the presence of a core hole [31], and the development of restricted open configuration interaction with single excitations (ROCIS) [32], all providing agreement with experimental XES results sufficient to enable detailed interpretations in terms of the orbitals involved in the transitions. However, while many of these methods are now very computationally cost-effective and resource-efficient, they can remain challenging from an experiential perspective for a large number of end-users, especially those new to the technique and those returning following its increase in popularity. Consequently, in parallel with the widening accessibility of next-generation, high-brilliance light sources putting the popularity of VtC-XES on an upward trajectory, it is important to develop accessible, affordable, and accurate tools for analysis.
In the present work, we build on our recent XANES-NET [33,34] deep neural network (DNN) for predicting the lineshapes of transition metal K- [33][34][35][36][37] and L-edge [38] XANES spectra to predict VtC-XES spectra. In contrast to recent work using unsupervised machine learning to invert spectra and directly extract chemically relevant information from them VtC-XES spectra [39,40] we focus on the forward (structure-to-spectrum) mapping. We investigate the performance of our network to predict VtC-XES across the first-row transition metal K-edges, and we show that -despite the increased sensitivity of VtC-XES to the electronic structure of the system under study -the VtC-XES DNN presented herein can reproduce the main spectral features from only the local coordination geometry of the transition metal complexes when encoded as a feature vector of weighted atom-centred symmetry functions (wACSF) [41], i.e. without explicit electronic information. We subsequently implement and evaluate three methods for assessing the uncertainty in the predictions made by the VtC-XES DNN, and demonstrate practical performance by application to unseen first-row transition metal (Ti-Zn) complexes. Table 1. Summary of the number of samples, N samples , in each of the nine first-row transition metal Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn reference datasets, the average energy of the onset for the VtC-XES spectrum, E calc , in eV, the X-ray absorption edge energies, E expt , in eV, and the width of the Lorentzian component, i , in eV used to broaden the calculated VtC-XES spectra with a Voigt function in a post-processing step (the Gaussian component used a fixed width of 1.5 eV) [42]. i reflects the core-hole lifetime broadening contribution as a sum of the 1s and 2p core-hole lifetime broadening. The difference, E, between E calc and E expt provides insight into the error in the calculated absolute transition energies, as such an error is well-documented for calculated XAS/XES spectra [43].   (Table 1).

Datasets
Our reference datasets comprise X-ray absorption site geometries ('samples') of first-row transition metal (Ti-Zn) complexes harvested from the transition metal Quantum Machine (tmQM) dataset [44,45]. The dataset for each first-row transition metal comprised all of the structures from the tmQM dataset containing that element, as extracted from the 2020 release of the Cambridge Structural Database (CSD) and subsequently optimised at the GFN2-xTB level of theory. This is described in detailed in ref. [45]. The tmQM dataset was initially generated by applying seven filters to exclude: (i) all structures except those containing a single transition metal; (ii) all structures not containing a minimum of one C and one H atom (allowing only these other elements: B, Si, N, P, As, O, S, Se, F, Cl, Br, and I); (iii) the structure of all extraneous molecular components beyond that of the transition metal complex (e.g. counter-ions); (iv) all polymeric structures; (v) all structures without threedimensional coordinates; (vi) all structures with disordered atoms; and (vii) all structures with charges greater than +1 and less than −1. Full details of the construction and composition of the tmQM dataset can be found in Ref. [45]. In addition to these constraints, our reference datasets further excluded structures containing Br and I. A summary of the number of samples contained in the reference datasets is given in Table 1. VtC-XES spectra ('labels') for all of the structures in our reference datasets were calculated using a quasi-oneelectron approach [15] implemented in the ORCA [46] quantum chemistry package. All VtC-XES spectrum calculations used the TPSSh [47,48] exchange and correlation density functional and the def2-SVP basis set [49]. The light-matter interaction was described using the electric dipole, magnetic dipole, and electric quadrupole transition moments [50]. After calculation, each VtC-XES spectrum was broadened using a Voigt function containing Lorentzian and Gaussian components. The Gaussian component reflects the limited experimental resolution of VtC-XES and had a fixed width of 1.5 eV in all cases. The Lorentzian component, i , reflects the effect of core-hole lifetime broadening and is a sum of the 1s and 2p core-hole lifetime broadening for each element, i.e. it is element-dependent. A summary of the value of i used with each first-row transition metal reference dataset is given in Table 1. Figure 1 shows the mean VtC-XES spectra and standard deviations for all VtC-XES spectra in each of the first-row transition metal reference datasets. A final pre-processing step was carried out to scale the target VtC-XES spectra for each transition metal reference dataset into the 0 → 1 interval independently by dividing through by the largest calculated cross-section in that reference dataset.
We have made the reference datasets publicly available (see our Data Availability Statement for details).

Deep neural network
The architecture of the VtC-XES DNN used in this Article closely follows that of XANESNET, as presented in Ref. [34] and is shown schematically in Figure 2. Briefly, it is a deep multilayer perceptron (MLP) model comprising an input layer, two hidden layers, and an output layer. All layers are dense, i.e. fully connected, and each hidden layer performs a nonlinear transformation using the hyperbolic tangent (tanh) activation function, shown as g() in Figure 2. The input layer comprises N neurons to accept a feature vector of length N encoding the local environment around the absorbing atom. This featurisation is performed via dimensionality reduction using the wACSF descriptor of Gastegger and Marquetand et al. [41]. In this Article, input layers containing either 49 or 50 neurons are used; in the former case, the feature vector comprises a global (G 1 ) function, 16 radial (G 2 ) functions, and 32 angular (G 4 ) functions and, in the latter case, the natural charge of the transition metal is additionally appended to the feature vector.
In the VtC-XES DNN used in this Article, the first hidden layer comprises 256 neurons with each subsequent hidden layer set to reduce in size by 10% relative to the size of the preceding layer. The output layer comprises 359 neurons from which the discretised K-edge VtC-XES spectrum is retrieved after regression. The architecture of the VtC-XES DNN used in this Article is consequently [N × 256 × 230 × 359]. The internal weights, W, are optimised via iterative feed-forward and backpropagation cycles to minimise the empirical loss, J(W), defined here as the mean-squared error (MSE) between the predicted, μ predict , and target, μ target , K-edge VtC-XES spectra over the reference dataset, i.e. an optimal set of internal weights, W * , is sought that satisfies argmin W (J(W)). The local geometries around first-row transition metal x-ray emission sites ('samples',) are inputs, and the corresponding theoretically calculated K-edge VtC XES spectra ('labels') are outputs. (b) The samples are encoded as descriptive feature vectors and associated with their labels to construct reference datasets from which the DNN discovers a 'forward' structure-to-spectrum mapping via iterative optimisation of the internal weights. The transformation occurring at each node including the outputs from the previous layer (x i ), the adjustable weights (w i ), non-linear hyperbolic tangent activation function (g) and output (y i ) is shown on in red.
Gradients of the empirical loss with respect to the internal weights, δJ(W)/δW, were estimated over minibatches of 32 samples and updated iteratively according to the Adaptive Moment Estimation (ADAM) [51] algorithm. The learning rate for the ADAM algorithm was set to 1 × 10 −4 . The internal weights were initially set according to the He [52] uniform distribution. Unless explicitly stated in this Article, optimisation was carried out over 2000 iterative cycles through the network commonly termed epochs. Regularization was implemented to minimise the propensity of overfitting; batch standardisation and dropout were applied at each hidden layer. The probability, p, of dropout was set to 0.20, unless otherwise stated.
The XANESNET DNN is programmed in Python 3 with the TensorFlow [53]/Keras [54] API and integrated into a Scikit-Learn [55] (sklearn) data pre-and postprocessing pipeline via the KerasRegressor wrapper for Scikit-Learn. The Atomic Simulation Environment [56] (ase) API is used to handle and manipulate molecular structures. The code is publicly available under the GNU Public License (GPLv3) on GitLab [57].

Uncertainty prediction
A level of uncertainty is unavoidable in machine learning models. In this Section, we evaluate strategies for estimating the confidence of the predictions that machine learning models make. Uncertainty within a machine learning model derives from two primary sources: (i) incomplete training data, e.g. where the training and testing data follow different distribution patterns (and, in which case, the machine learning model may be unequipped to describe the feature space spanned by the testing dataset), and (ii) model uncertainty, e.g. where there is no guarantee that there exists a single unique solution to the problem of finding a set of internal weights, W * , which satisfy argmin W (J(W)), and reoptimising the model multiple times may yield different W * .
We adopt three approaches to estimate the uncertainty in the VtC-XES DNN used in this Article: (a) ensembling, (b) Monte-Carlo dropout, and (c) bootstrap resampling. The approach and motivation for each these are described in Subsections (a) 2.3.1, (b) 2.3.2, and (c) 2.3.3 below, and each is illustrated schematically in Figure 3.

Ensembling
The initial weights in the VtC-XES DNN are drawn from the He [52] uniform distribution (Section 2.2), and there is no guarantee that there exists a single unique solution to the problem of finding a set of internal weights, W * , which satisfy argmin W (J(W)). An ensemble of statistically-initialised VtC-XES DNNs optimised using the same reference dataset can consequently yield different W * . Ensembling (Figure 3(a)) involves optimising multiple instances of a machine learning model with the same reference dataset but with a different statistical initialisation of the internal weights. The ensemble of N machine learning models is then used to produce N independent predictions for each sample in the 'held-out' testing dataset from which a mean prediction and standard deviation for each sample can be derived. The latter quantifies the ensemble uncertainty. The concept behind this interpretation is that different W * obtained from the N models in the ensemble will tend towards similar spectral predictions when the structural inputs are comparable to those used in the training data. This is because each instance's weights, even if different, are optimised for comparable data. In contrast, if the structural inputs are dissimilar to the structures used to train the model, each of the N independent predictions will be more affected by the specificities of W * , thus the higher standard deviation will be observed.

Monte-Carlo dropout
The dropout regularisation technique (in which neurons are probabilistically 'turned off' and 'dropped out' of the weight update cycle with each feedforward/backpropagation epoch) is commonly used during optimisation to reduce model complexity and the propensity for overfitting [58]. The Monte-Carlo dropout technique (Figure 3(b)) additionally applies this probabilistic dropout at inference, or 'prediction' time, resulting in different outputs for the same input if the machine learning model is used repeatedly in inference mode. From N independent predictions with probabilistic dropout at inference time (analogous to sampling over N different DNN configurations), a mean prediction and standard deviation for each sample in the 'held-out' testing dataset can be derived. The latter, as in the ensembling approach (Section 2.3.1) quantifies a 'dropout-configurational' uncertainty which is related to a stochastic realisation of the Bayesian estimation for model uncertainty [59,60]. The effect of Monte-Carlo dropout quantifies the similarity of the input sample with the samples in the training dataset; the basis for this is that a higher or lower similarity will lead to a prediction which is less or more greatly affected by the dropout applied at inference time, respectively. A drawback of Monte-Carlo dropout is in its dependence on the hyperparametric dropout rate, p (Section 2.2), as the choice can have an effect on both on the accuracy of the machine learning model and the uncertainty estimation [58].

Bootstrap resampling
The bootstrap resampling technique (Figure 3(c)) is used to estimate statistics on a population by sampling a dataset with replacement, and can be used to describe the uncertainty associated with incomplete training data [61,62]. N machine learning models are optimised using N reference datasets sampled with replacement from  the original reference dataset; each one of these is the same size as the original reference dataset and, consequently, may contain repeated instances of the same sample. The bootstrap resampling technique provides increased dataset diversity to each instance of the machine learning model. N independent instances of the machine learning model optimised using N bootstrapped reference datasets are then used to produce N independent predictions for each sample in the 'heldout' testing dataset from which a mean prediction and standard deviation for each sample can be derived. In contrast to ensembling (Section 2.3.1) and Monte-Carlo dropout (Section 2.3.2), bootstrap resampling relies on injecting diversity into the reference dataset, rather than sampling diverse machine learning model configurations.   Figure 4) show the MSE where the natural charge of the transition metal is exposed to the VtC-XES DNN explicitly via appending an additional feature to the feature vector input (Section 2.2), while the grey boxes (excl. charges; Figure  4) show the MSE where only the local geometry around the X-ray emission site, i.e. the feature vector, is input. The counterpart plots for each of the other transition metal reference datasets (Ti-Zn) are presented in Figures S1-S8. For all of the transition metal reference datasets used in this Article, the VtC-XES DNN can be optimised to convergence in ∼ 1000-2500 feedforward/backpropagation epochs.

Optimisation and performance
These results show that although including the natural charge of transition metal as an additional input yields a marginally lower MSE in the many-epoch limit, it does not have a substantial effect on the convergence or the predicative power of the VtC-XES DNN either at the Mn Kβ 2,5 -edge (Figure 4) or at any of the other transition metal Kβ 2,5 -edges Figures S1-S8. As a VtC-XES DNN relying on this additional information would require the natural charge on the transition metal to be computed in a quantum-chemical calculation prior to prediction of the VtC-XES spectrum for an unknown sample (an additional step that the objective of fast, affordable, and accessible predictions of VtC-XES spectra necessitates is avoided; Section 1), the advantages of the machine learning model would be limited were it required. Consequently, although we do not rule out the possibility of circumventing these challenges by supplying machine-learned natural charges available through another machine learning model, we continue here in this Article without providing information on the natural charge explicitly, i.e. by providing only the local geometric information as input to the VtC-XES DNN as a feature vector of a global G 1 , 16 G 2 and 32 G 4 wACSF (Section 2.2). Figure 5 shows the median percentage error, μ (E) median , between μ predict and μ target for each of the first-row transition metal 'held-out' testing datasets as a function of energy; for comparison, the counterpart plots where the natural charge on the transition metal is included are presented in Figure S9. μ(E) median is typically < 15% at all discretised points over the spectral energy window and for all of the first-row transition metal reference datasets; it is only greater in regions where the mean VtC-XES spectral intensity ( Figure 5; light blue trace) is near-zero, i.e. where negligible inaccuracies in comparatively unimportant windows of the VtC-XES spectrum are able to give rise to a spuriously large percentage error. Figure 6 shows a box plot of the μ distribution for each of the first-row transition metal 'held-out' testing datasets; complementary summary statistics are tabulated in Tables S1 and S2. The median μ over the testing dataset, μ median , is typically ca. 20% with the lower and upper quartiles situated symmetrically ca. 3-8% under and above, respectively, giving an interquartile range (IQR) of ca. 6-18%. Coupled with the moderate positive skewness coefficients (ca. 1-2%; Tables S1 and S2), predictions from the VtC-XES DNN are typically placed towards the higher-performance end of the μ distribution. μ -as determined here for the VtC-XES DNN -is considerably larger than reported for the XANESNET DNN [33,34] in the XANES domain at the first-row transition metal K-edges ( μ median is typically ca. 5% with an IQR of ca. 3-5% with larger skewness coefficients of ca. 2-3%; Ref. [34]) but, as seen in Figure  5 and discussed in the preceding paragraph, a significant contributor to the higher percentage error realised here is that there are regions of the VtC-XES spectrum where μ target is near-zero and negligible -even numeric -inaccuracies here are able to give rise to large percentage errors.

Uncertainty prediction
Having reviewed the performance of the VtC-XES DNN, we turn our attention to a different question: is it possible to predict, given the input sample, whether the VtC-XES DNN will fail/produce an unreliable predication? Table 2 compares properties of the μ distribution, obtained for VtC-XES DNN predictions on the contents of the 'held-out' reference datasets and ranked by mean squared error. The properties calculated include μ median , the upper and lower quartiles, the median standard deviation, σ median , and the skewness coefficient. These have been obtained using the ensembling (Section 2.3.1), Monte-Carlo dropout (Section 2.3.2), and bootstrapping (Section 2.3.3) approaches. We also introduce another metric into Table 2 -percentage coverage -which quantifies the percentage of discretised points over the whole spectral energy window for which μ target is within the region bounded by μ predict ± 2σ .
The μ median , upper and lower quartiles, σ median and the skewness coefficient are comparable to those found in the previous section (see Tables S1 and S2) meaning that the implementation of the ensembling, Monte-Carlo dropout and bootstrapping methods does not alter the underlying performance pf the VtC-DNN, rather simply provides an appraoch for potentially estimating the uncertainty. For this we observed that the Monte-Carlo dropout yields the largest percentage coverage over all of the first-row transition metal datasets, indicating that approach can provide the most confidence in the predictions, but, as shown in Table 2, this is primarily a consequence of the larger standard deviation that the approach generates. Indeed, for the ensembling and bootstrap resampling techniques, we find that the Table 2. Summary of the median percentage errors, μ median (%), upper and lower quartiles, skewness coefficients, median standard deviations, σ median , and percentage coverage (%) for the μ distributions. standard deviations are very similar (0.0038 and 0.0033, respectively), with the Monte-Carlo approach yielding a larger standard deviation of 0.0052. Overall, on the basis of the numerical analysis there is very little different between the three models and all have little effect on the underlying predicted performance of the networks and so we now turn to their ability to predict the uncertainty associated with that predicted VtC-XES spectrum. Figure 7 shows a scatter plot of μ median against where i represents each discretised spectral energy point in the spectrum, for bootstrap-resampled VtC-XES DNN predictions of the contents of the training (plot in light grey) and 'heldout' testing (plot in black) data for each of the first-row transition metal reference datasets. The Pearson correlation coefficients, ρ (inset in Figure 7), indicate moderate association across all of the reference datasets. The corresponding plots for ensembling and Monte-Carlo dropout are shown in Figures S10 and S11. This moderate association shown in Figure 7 suggests that the value of Norm. σ extracted from a bootstrap-resampled VtC-XES DNN prediction might be useful as a crude indicator of the potential uncertainty associated with that predicted VtC-XES spectrum. Importantly, the association is significantly weaker for both the ensembling and Monte-Carlo dropout approaches, as these reflect sampling diverse machine learning models in contrast to the bootstrap resampling, which relies on diversity into the reference dataset. This suggests that the main source of uncertainty is associated with the composition of the reference dataset, however, future work should investigated this in more detail.

2.3.3)
, i.e. they are a proxy for the potential uncertainty associated with that predicted VtC-XES spectrum (Section 3.2). The space bounded by ±2σ is greatest for the weakest-performing predictions in the 90-100th percentile (Figure 8; lower panels) and smallest for the best-performing predictions in the 0-10th percentile ( Figure 8; upper panels), indicating increased confidence of the VtC-XES DNN on the most accurate predictions and reduced confidence on the least accurate predictions, i.e. reinforcing the relationship presented in Figure 7. All VtC-XES spectral predictions exemplified in Figures 8 and S12-S19 reproduce the characteristic Kβ 2,5 and weaker Kβ transitions located at lower energies: transitions that are associated with electrons relaxing from ligand p and s orbitals to fill the 1s core-hole on the transition metal. [21] A consequence of this association is that the intensity and the lineshape of the transition is a sensitive probe of the type of ligand. Among the weakest predictive performances given by the VtC-XES DNN for the Mn testing dataset are trans-bis(azido)tetraazacyclotetradecane manganese(III) (CSD code: SIQLOT), tris(1,2-bis(dimethylphosphino)ethane) manganese(I) (CSD code: BETQAT), and bis((1,2-bis(dime thylphosphino)ethane)-(phenylethynyl)) manganese(I) (CSD code: TEBXIH) -all of which belong to the 90-100 th percentile when the VtC-XES spectral predictions are ranked by performance (Figure 8; lower panel). These complexes (and others with weak predictive performance, i.e. large μ and σ median ) contain either rare ligands within the context of the reference training dataset, e.g. azides (SIQLOT), or ligands binding through elements that are relatively underrepresented in the reference training dataset, e.g. P (BETQAT and TEBXIH).

Discussions and conclusions
In this Article, we have extended our XANESNET DNN [33,34] to develop a VtC-XES DNN that is able to predict the lineshapes of first-row transition metal (Ti-Zn) Kβ 2,5 VtC-XES spectra. After experimenting with three techniques (ensembling, Monte-Carlo dropout, and bootstrap resampling) to quantify the uncertainty on the VtC-XES spectral predictions made by the VtC-XES DNN, we have ultimately chosen to implement the bootstrapping resampling technique and, through doing so, developed an uncertainty metric which can be used to estimate the reliability of the VtC-XES spectral predictions and the confidence of the VtC-XES DNN.
The models developed are able to predict K-edge VtC XES spectral intensities with an average accuracy of ca.
20% across the selected spectral windows. This is slightly larger than previously reported for K-edge XANES spectra [34], however a significant contributor to the higher percentage error is that there are regions of the VtC-XES spectrum where μ target is near-zero and negligible -even numeric -inaccuracies here are able to give rise to large percentage errors. It should also be stressed that in contrast to XANES spectra studied previously, the VtC XES are dominated by electronic transitions, in this case one-electron transitions from the valence orbitals to the core orbitals. As our descriptor is purely based upon structural information, this could contribute to the larger error as highlighted in previous work at the Pt L-edge [38]. At the transition metal K edge, the majority of the features in the XANES spectrum are above-ionisation resonances which arise through interferences of the scattered X-ray photoelectrons and, here, the direct link to between structure and spectrum (necessary for a physical 'forward' mapping) is clear. In contrast, at the VtC X-ray Emission spectra, there is a substantially larger contribution from electronic characteristics and, while there is still an implicit link between the geometric properties of the local environment around the absorption site and the electronic structure of the complex, it is not as direct. While we incorporated some of the information required through the natural charge of the metal ion, this does not have sufficient benefit to warrant the complication it adds to the network.
The Bootstrapping approach has proved successful in estimating the uncertainty expected between the target and prediction spectrum. The Pearson correlations show moderate to strong association providing an encouraging starting point to be able to quantify the quality of the predictions. Importantly, the association is significantly weaker for both the ensembling and Monte-Carlo dropout approaches. As these appraoches reflect sampling diverse machine learning models, in contrast to the bootstrap resampling, which relies on diversity into the reference dataset, it suggests that the main source of uncertainty is associated with the composition of the reference dataset. The uncertainty metric also gives rise to another dimension, active learning. At present, our datasets have been developed simply extracting all the of samples found within the tmQM training set, this is unlikely to be optimised for VtC XES. However, using the uncertainty gained from the predictions one could select structures to add to the training set in an active learning manner.