Inverting the Kohn–Sham equations with physics-informed machine learning

In the framework of KS-DFT, we solve a set of effective single-particle equations known as the KS equations (which we present here using atomic units [77] for clarity and simplicity of notation),

$$\begin{align} \left[ -\frac{\nabla^2}{2} + v_{s}\left(\boldsymbol{r}\right) \right] \phi_i\left(\boldsymbol{r}\right) = \epsilon_i ~ \phi_i\left(\boldsymbol{r}\right)\,, \end{align} \tag{ 1 }$$

where $\phi_i(\boldsymbol{r})$ represents the KS eigenfunctions and ε_i denotes the KS eigenvalues. These KS equations constitute an auxiliary system designed to replicate the electron density of an interacting many-body system, expressed as

$$\begin{align} n\left(\boldsymbol{r}\right) = \sum_{i = 1}^{N} \vert \phi_i\left(\boldsymbol{r}\right) \vert^2\,, \end{align} \tag{ 2 }$$

where the summation index corresponds to the total number of electrons, denoted as N.

The correspondence of the density defined in equation (2) with the density of an interacting many-body system is established through the KS potential which mimics the inter-electronic interaction and is defined as

$$\begin{align} v_\mathrm{{\scriptscriptstyle S}}\left(\boldsymbol{r}\right) = v_\mathrm{{\scriptscriptstyle ext}}\left(\boldsymbol{r}\right) + v_\mathrm{{\scriptscriptstyle H}}\left(\boldsymbol{r}\right) + v_\mathrm{{\scriptscriptstyle XC}}\left(\boldsymbol{r}\right)\ . \end{align} \tag{ 3 }$$

Here, $v_\mathrm{{\scriptscriptstyle ext}}(\boldsymbol{r})$ is the external potential, $v_\mathrm{{\scriptscriptstyle H}}(\boldsymbol{r})$ is the Hartree potential (i.e. the electrostatic potential due to an electronic density interacting with itself) and $v_\mathrm{{\scriptscriptstyle XC}}(\boldsymbol{r})$ is the XC potential. Both the external and Hartree potentials are known exactly in terms of the density. The only unknown is the XC potential. While solving the set of KS equations with the exact XC potential yields the exact density of the many-body system, in practice, approximations to the XC energy need to be used. A plethora of approximations is available in the literature [2, 4–13] and more accurate approximations are continuously being developed.

Usually, the KS equations are solved iteratively. This means that a system is defined and an XC approximation is selected. An initial guess of the ground-state density is made, which in turn provides guesses for the Hartree and XC potentials. Equations (1) and (2) are solved, generating a new density, which can be used to calculate new Hartree and XC potentials. This cycle will continue until the density converges to within a chosen criterion. In the end, the resulting density is an accurate approximation of the interacting many-body system, which can be used to calculate electronic properties.

Conversely, equations (1) and (2) can be solved to find the exact KS potential that results in a given target density. In this case, no approximation is made to the XC potential, only an initial guess. Then, by solving equations (1) and (2), the guess for the XC potential can be updated. This process is much less stable than the 'direct' one described above because the potential is sensitive to small changes in the density.

We use the iDEA code [78, 79] for generating data and as benchmark for inversion. In iDEA, the density-to-potential inversion is done iteratively:

$$\begin{align} v_{s} \rightarrow v_{s} + \mu \left[n\left(\boldsymbol{r}\right)^p - n_0\left(\boldsymbol{r}\right)^p\right]\,, \end{align} \tag{ 4 }$$

where $n_0(r)$ represents the target density, while µ and p serve as convergence parameters.

Model systems in one dimension are a useful tool for the development of DFT approximations [59] for two reasons. First, it is simpler to obtain highly accurate or exact data from a one-dimensional system. Second, these model systems can be finely tuned to mimic a wide range of well-understood physical phenomena. Therefore, we leverage one-dimensional models to demonstrate the utility of our inversion method. We define two model systems: a one-dimensional atom and a one-dimensional diatomic molecule, as illustrated in figure 1. A one-dimensional atom is defined by the external potential

$$\begin{align} v_\mathrm{{\scriptscriptstyle ext}}\left(\boldsymbol{r}\right) = -\frac{Z}{|\boldsymbol{r}|+a}\,, \end{align} \tag{ 5 }$$

where Z is the charge of the atomic well and a is softening parameter to ensure that the potential is defined at all points in space. This simple model can readily be extended to a one-dimensional model of a diatomic molecule by incorporating two potential wells through an additional parameter, d, which represents the bond length in terms of the separation distance between the two wells:

$$\begin{align} v_\mathrm{{\scriptscriptstyle ext}}\left(\boldsymbol{r}\right) = -\frac{Z_1}{|\boldsymbol{r}-\frac{d}{2}|+a} - \frac{Z_2}{|\boldsymbol{r}+\frac{d}{2}|+a}. \end{align} \tag{ 6 }$$

This idea can be extended by defining additional wells to move from an atomistic view to a material or crystalline structure in one dimension.

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All data used to train the presented models were generated using external potentials defined in this manner. The electronic densities and KS eigenfunctions were generated using the iDEA code [78, 79]. For the one-dimensional atom of varying charge, Z was varied between the values of two and six in steps of 0.1, resulting in a dataset with 41 individual data points which was split into a train, test, and validation set. These sets had 32, 5, and 4 individual data points each, respectively. For the one-dimensional diatomic molecule, Z₁, Z₂, and d were all varied within the range of one to five, in steps of 0.5. All combinations of the three variables were generated, resulting in a data set with 729 data points. The dataset was split such that the training set had 590 points, the test set had 73, and the validation set had the remaining 66. The dataset can be downloaded from the HZDR Data Repository Rodare via the link https://rodare.hzdr.de/record/2720 [80].

As a subset of neural networks, PINNs use equations and conditions of the underlying physics to enable a neural network to predict physically relevant information. Consider a general partial differential equation of the form

$$\begin{align} u\left(x,t\right) + \mathcal{N}\left[u;\lambda\right] = 0\,, \end{align} \tag{ 7 }$$

where u is the solution, $x \in \Omega$ a D-dimensional spatial coordinate with Ω a subset of $\mathbb{R}^D$, $t \in [0,T]$ the time variable, $\mathcal{N}$ a nonlinear operator, and λ the set of parameters that describe a physical system. If a neural network is used to represent the solution u, as $\tilde{u}$, this gives rise to

$$\begin{align} \tilde{u}\left(x,t\right) + \mathcal{N}\left[\tilde{u};\lambda\right] : = f\left(x,t\right)\ . \end{align} \tag{ 8 }$$

The function f must only equal zero when the network accurately represents the solution to the differential equation for the given parameters λ. If trained on a wide set of physical data, it should be able to predict solutions to the differential equation within that set more generally.

Based on equation (1), we derive the following model and loss functions within the framework of PINNs:

$$\begin{align} f\left(\boldsymbol{r}\right)&=\epsilon_i ~ \phi_i\left(\boldsymbol{r}\right)+\frac{\nabla^2}{2}~\phi_i\left(\boldsymbol{r}\right)- v_{s}\left(\boldsymbol{r}\right)~\phi_i\left(\boldsymbol{r}\right) \end{align} \tag{ 9 }$$

$$\begin{align} L_\mathrm{{\scriptscriptstyle PDE}} &= \mathrm{}{MSE}\left(f,0\right) = \frac{1}{N_f} \sum_{i = 1}^{N_f}\left| f\left(x_f^i\right) \right|^2\,, \end{align} \tag{ 10 }$$

where the loss function $L_\mathrm{{\scriptscriptstyle PDE}}$ is computed in terms of a mean squared error (MSE) $\mathrm{}{MSE}(f,0)$, with $x_f^i$ denoting one of the N_f collocation points where the solution is provided.

Here, our focus is to invert the KS equation, wherein we have the set of solutions, $\{\phi_i\}$, and our goal is to get the network to predict the parameters, $v_\mathrm{{\scriptscriptstyle S}}$, that lead to an appropriate set of solutions. If the network can correctly predict $v_\mathrm{{\scriptscriptstyle S}}$, then the function f should return a value of zero. What distinguishes this approach from conventional data-driven ML is that the network does not receive the solution set as input for prediction. Instead, it receives only the weighted sum of the solutions in the form of the electronic density calculated using equation (2).

In figure 2, we illustrate our implementation of a PINN that predicts the KS potential for an input density. The network takes the density as an input and processes it through a series of convolutions. The output of each convolution as well as the initial density is used as an input into a dense network. The network is then asked to predict the KS potential on each point in space given this set of features, as well as the first and second derivative of the density.

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Convolutional neural networks are a common ML technique for image recognition [81–84]. They observe both local and non-local features in input data to predict what is present in an image. The convolution of two functions f and g is defined as

$$\begin{align} \left(f*g\right)\left(t\right) = \int f\left(\tau\right)~g\left(t-\tau\right)~d\tau \end{align} \tag{ 11 }$$

and can be described as sum of the overlap of two functions at each point in t. In our convolutional PINN, this is accomplished by using a set of convolutions in series and in parallel. Back propagation is used to learn functions g that emphasize input features crucial for predicting a class. Increasing the number of convolutions in series results in the passing of additional non-local information to the features.

It is widely known [1, 4] that the XC potential is a nonlocal functional of the electronic density. Consequently, training a network to correctly predict an XC potential based only on the electronic density at a single point in space is ill-advised. Our PINN model uses convolutions to account for some of the non-locality of the XC potential. Each step through the convolutions indicates a larger regime of non-locality that has been included in its output features.

Neural operators (NOs) [85, 86] are an extension of neural networks that take functions as inputs and return functions as outputs. While traditional neural networks map finite-dimensional vectors to finite-dimensional vectors, neural operators work on infinite-dimensional function spaces. A neural network can be defined as a mapping $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, whereas a neural operator maps functions to functions, i.e. $G: \mathcal{A} \rightarrow \mathcal{U}$, where $\mathcal{A}$ and $\mathcal{U}$ are function spaces. Given that there exists a non-linear map $\mathcal{G}^{\dagger}: \mathcal{A} \rightarrow \mathcal{U}$, our goal is to construct a neural operator that approximates this map, i.e. $\mathcal{G}_{\theta} \approx \mathcal{G}^{\dagger}$, where $\mathcal{G}_\theta: \mathcal{A} \rightarrow \mathcal{U}$ with parameters θ in finite dimensional space $\mathbb{R}^p$.

The neural operator is trained on point-wise (in function space) observations of the form $\{(a_i, u_i) \}_{i = 1}^N$, where $\left( a_i \in \mathcal{A}; a_i(x) \in \mathbb{R}, x \in \mathbb{R} \right)$, $\left( u_i \in \mathcal{U}; u_i(x) \in \mathbb{R}, x \in \mathbb{R} \right)$ and $u_i = \mathcal{G^{\dagger}} (a_i)$. The goal is to find the parameters $\theta^*$ that minimize the loss

$$\begin{align} \theta^* = \min _{\theta \in \mathbb{R}^p} \frac{1}{N} \sum_{i = 1}^N\left\|u^{\left(i\right)}-\mathcal{G}_\theta\left(a_i\right)\right\|_{\mathcal{U}}^2. \end{align} \tag{ 12 }$$

Analogous to the neural networks being defined as multi-layer compositions of linear and non-linear operations, we can define a neural operator $\mathcal{G}_\theta(a)$ as multi-layer compositions of linear and non-linear operators acting directly in function space:

$$\begin{align} \mathcal{G}_\theta\left(a\right) = Q\left(v_L\left(v_{L-1}\left(\dots v_1\left(P\left(a\right)\right)\right)\right)\right)\,, \end{align} \tag{ 13 }$$

where the layer $v_{l+1}$ is defined as

$$\begin{align} v_{l+1}\left(x\right) = \sigma_{\left(l+1\right)}\left(W_l v_l\left(x\right)+\left(\mathcal{K}_l\left(a ; \lambda\right) v_l\right)\left(x\right)\right)\ . \end{align} \tag{ 14 }$$

Here, the non-local kernel integral operator $\mathcal{K}_l$ is defined as

$$\begin{align} \left(\mathcal{K}_l\left(a ; \lambda\right) v_l\right)\left(x\right) = \int_{\Omega_{l}} \kappa_{l}\left(x, y,a\left(x\right), a\left(y\right)\right) v_l\left(y\right) d y \end{align} \tag{ 15 }$$

over the domain $\Omega_l$.

The kernel function $\kappa_l(x,y,a(x), a(y))$ is a function of $x,y$, and the input functions a(x) and a(y). The kernel function, parametrized by λ, and the local linear operator W_l , a matrix of weights, are learned during training. $\sigma_{(l+1)}$ is a local non-linear activation function. The local operator P(a) is a pre-processing operator that transforms the input function a into a higher dimensional representation v₀, which is the input to the first layer. The local operator $Q(v_L)$ is a post-processing operator that projects the output of the last layer v_L back into $\mathcal{U}$.

Several methods can be used to approximate the global integral calculation in each layer, such as graph neural networks with Monte-Carlo sampling [86] and multipole graph neural operators [87]. These approximations reduce the computational complexity and allow for scalable solutions [76].

FNOs [72] leverage the Fourier transform to work in the frequency domain. Notice that by using a kernel $\kappa(x-y)$, equation (15) is a convolution operator, and by using the convolution theorem, equation (15) becomes

$$\begin{align} \left(\mathcal{K}_l\left(a ; \lambda\right) v_l\right)\left(x\right) &= \mathcal{F}^{-1}\left(\mathcal{F}\left(\kappa_l\right) \cdot \mathcal{F}\left(v_l\right)\right)\left(x\right) \end{align} \tag{ 16 }$$

$$\begin{align} \mathcal{F}\left(x\right) &= \sum_{n = 0}^{N-1}~x_n \cdot e^{-\frac{i2\pi}{N}kn} = X_k \end{align} \tag{ 17 }$$

$$\begin{align} \mathcal{F}^{-1}\left(X\right) &= \frac{1}{N}\sum_{n = 0}^{N-1}~X_k \cdot e^{\frac{i2\pi}{N}kn} = x_n, \end{align} \tag{ 18 }$$

where $\mathcal{F}$ is the Fourier transform and $\mathcal{F}^{-1}$ is its inverse. The kernel $\kappa_l$ is parametrized in Fourier space. Here the parameterization is finite-dimensional determined by the number of Fourier modes $k_\textrm{max}$. By working in frequency domain, the kernels can be computed efficiently using the fast-Fourier tranform and the model can capture global patterns more effectively.

A distinction has to be made between the solving approach (PINNs) and the learning approach (NOs) for PDE problems. PINNs are useful for obtaining numerical solutions for specific initial and boundary conditions and PDE parametrizations. In contrast, learning a PDE via neural operators aims to learn the underlying operator itself, enabling the prediction of solutions for various conditions without re-solving the PDE. PINNs can be used to solve PDEs if the PDEs are defined, even in the absence of training data, while NOs can be used to approximate undefined PDEs with training data. NOs are resolution-invariant and can also be used for zero-shot super-resolution [72]. A rigorous mathematical definition and analysis for NOs is provided in [76]. The strengths of FNOs in this context lie in their inherent capability to capture non-local dependencies and their resolution invariance.

Given the non-local dependence of the XC potential on the electronic density, the non-locality of FNO layers make them well-suited for the inverse problem in KS-DFT. The input in this case is a grid of electronic density values and the output is the XC potential at the corresponding spatial grid points. The loss function is simply the MSE between true and predicted XC potential:

$$\begin{align} L_\textrm{FNO} = \mathrm{}\textrm{MSE}\left(v_\mathrm{{\scriptscriptstyle XC}}, \tilde{v}_\mathrm{{\scriptscriptstyle XC}}\right) = \frac{1}{N} \sum_{i = 1}^{N}\left| v^{\left(i\right)}_\mathrm{{\scriptscriptstyle XC}} - \tilde{v}^{\left(i\right)}_\mathrm{{\scriptscriptstyle XC}}\right|^2\, \end{align} \tag{ 19 }$$

where $\tilde{v}_\mathrm{{\scriptscriptstyle XC}}$ denotes the predicted XC potential for system i. The architecture we implemented is shown in figure 3.

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We evaluate the complexity of the datasets using two different similarity measures, namely the cosine similarity and Euclidean distance. The cosine similarity between two vectors X and Y in $\mathbb{R}^D$ is defined as

$$\begin{align} \text{cos}\left(\theta\right) = \frac{\mathbf{X} \cdot \mathbf{Y}}{\Vert \mathbf{X} \Vert \Vert \mathbf{Y} \Vert}, \end{align} \tag{ 20 }$$

which returns values in the range $[-1,1]$, namely −1 if the vectors point in opposite directions, 1 if they point in the same direction, and 0 if they are orthogonal to each other. The Euclidean distance between the same vectors is defined as

$$\begin{align} D\left( \mathbf{X}, \mathbf{Y} \right) = \Vert \mathbf{X} - \mathbf{Y} \Vert. \end{align} \tag{ 21 }$$

This measure has a lower bound of zero but no upper limit. The value returned is 0 only if the vectors X and Y are the same vector. Then, as two vectors continue to move further apart, the value moves further away from zero. The Euclidean distance includes some details related to the cosine similarity, but it also describes the difference in magnitude of the values found in the two vectors. This is a significant piece of information that the cosine similarity measure misses. Both of these measures provide insight into the complexity of each dataset and the network architecture's capacity to work on each of them.

We illustrate the similarity for both the atomic and the molecular model systems in figure 4. As expected, the diatomic system is more complex than the atomic system, which is evident from the broader range of values in the diatomic case, versus the atomic case, for both cosine similarity and Euclidean distance measures. In the cosine similarity plots for the diatomic system, a block pattern emerges. The largest blocks shift with a change in the bond length. Descending to the bottom or moving to the right of the plot corresponds to an increase in the separation between two atomic centers. The greatest dissimilarity is noted at the two extremes of separation. Interestingly, at larger separations, the vectors show slightly less self-similarity compared to those at lower separations, as indicated by the darker shading of the bottom right block compared to the top left. The Euclidean distance plot yields similar trends, although it does not show this distinction within varying separation distances.

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3.1.1. Atom with varying charge

Figure 5 illustrates the performance of the PINN model for a one-dimensional atom with a variable parameter Z corresponding to its charge, plotting the MAE for a set of fixed charges. Each data point represents the mean of ten neural network predictions that were generated using varying network initializations and random seeds. The data points are color-coded to indicate ground and excited states in the order of blue, orange, red, green, purple, and brown. This order is maintained in the following discussion of the results.

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The performance of the PINN model is best when the well charge Z is small. The error systematically increases with the charge, particularly for the lowest-lying eigenvalue. The overall MAE averaged over the test systems is 1.77$\times\,10^{-3}$ Ha (denoted by a gray horizontal line in figure 5). The PINN model is competitive with the chemical accuracy criterion of being within 1 kcal mol⁻¹ or 0.0016 Ha, as marked by the red horizontal line in our figures.

We have also displayed the the MAPE of the individual eigenvalues in figure 6. The greatest error is in the first two eigenvalues of the systems, with both lying above the total average MAPE of around 0.13%, while the next four lie below this threshold.

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We have also explored the relationship between the performance of the PINN model and the resolution of the spatial grid. We suspect that the source of the errors observed in figure 5 is related to the resolution of the grid used to train the PINN model, which depends on the accuracy of the KS orbitals and their second derivatives. We anticipate that as the grid resolution improves, the accuracy of the second derivatives will also improve, particularly at the sharp cusp-like feature in the KS potential. Our findings indicate that errors diminish with the enhancement of grid resolution. This correlation presents a methodical approach for reducing the error in the PINN model. The improvements are reported in table 1, which showcases the systematic enhancement in performance with increasing grid resolution (see Atom 501).

Table 1. Performance comparison of PINN and FNO models on both model systems, showing the MAE (in Ha) and MAPE (%) and denoting the maximum absolute errors and the maximum percentage errors in braces. The labels 301 and 501 indicate the grid resolutions and E denotes extrapolation.

Model	Atom (301)	Atom (501)	Molecule (301)	Molecule (301 E)
PINN	0.0018 Ha	0.0007 Ha	0.0125 Ha	0.0248 Ha
	(0.0110 Ha)	(0.0034 Ha)	(0.1000 Ha)	(0.1200 Ha)
	0.13%	0.06%	0.58%	0.73%
	(0.30 %)	(0.15%)	(2.37%)	(3.25%)
FNO	0.0002 Ha	0.0002 Ha	0.0002 Ha	0.0054 Ha
	(0.0009 Ha)	(0.0010 Ha)	(0.0051 Ha)	(0.0584 Ha)
	0.03%	0.03%	0.02%	0.25%
	(0.15%)	(0.19%)	(0.46%)	(3.51%)

3.1.2. Diatomic molecule with varying charges and bond length

When switching to the diatomic molecule, the learning task becomes much more complex, as both the charges (Z₁ and Z₂) and the bond length (d) are variable parameters. This is exemplified in the roughly 10-fold increase in errors within the predicted eigenvalues. The MAE is shown in figure 7. Compared to the atomic test system, which had an overall MAE averaging around 1.77 $\times\,10^{-3}$ Ha, the overall MAE has increased to 1.25 $\times\,10^{-2}$ Ha (refer to the gray horizontal line in figure 7). In the atomic test system, the largest errors were observed in the ground, first, and second states of the KS spectrum. All eigenvalues exhibit some errors above the limit of chemical accuracy. This suggests that the PINN model has difficulty predicting the tails of the XC potential, but not to the same extent as placing and accurately estimating the well depth. The relative ordering of errors on the states remains consistent.

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We have further broken down the error analysis into individual eigenvalues in figure 8. We display the MAPE for the first six eigenvalues, which exhibits the larger errors for the ground and first excited states observed in the MAEs plot. The overall MAPE averages around 0.575% (denoted by a gray horizontal line in figure 8).

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3.1.3. Extrapolation for diatomic molecule

Purely data-driven neural network approaches rapidly degrade in predictive performance when provided with data outside the training set. However, using physics-based approaches such as PINNs and also FNOs, higher extrapolation performance can be expected. Compared to data-driven learning, the neural network in the PINN model represents the solution to a given PDE, allowing accurate predictions within the specific scope of that PDE. We evaluate the PINN model's extrapolation ability by asking it to predict on a set of densities outside the training set.

The outcome of this analysis is illustrated in figure 9, which shows the MAE. The errors have increased as anticipated, resulting in an overall MAE of 2.48 $\times\,10^{-2}$ Ha indicated by the gray horizontal line. The error has doubled in comparison to the interpolation task featured in figure 5. While this is not a catastrophic failure, the higher error suggests the intricacy of extrapolation. The challenge of extrapolation is evident in figure 10, where we illustrate the MAPE resolved over the first six eigenvalues with an overall MAPE of 0.73%, denoted by a gray line. Although no significant or sudden rise in error is observed with increasing eigenvalues, the overall MAPE is noticeably higher, as is the standard deviation shown by the red error bars.

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In contrast to the PINN model, the density-to-potential mapping data is explicitly used in the loss function to train the data-driven FNOs, as outlined in section 2.4. The network takes the density grid as input and produces the corresponding potential grid as output. FNOs have several benefits, including learning the mapping in the functional space, which enables them to generalize effectively over varying input densities. Additionally, the network is resolution-invariant, allowing it to assess higher resolution grids than those present in the training dataset.

3.2.1. Atom with varying charge

In the case of the one-dimensional atom with varying charge Z, the FNO model demonstrates exceptional performance over the range of charge values. The error plots for the eigenvalues, as shown in figure 11, indicate that the predicted eigenvalues closely align with the true values. We resolve the error analysis over the spectrum of eigenvalues by showing the MAPE for the first six eigenvalues in figure 12. The red error bars denote three standard deviations away from the mean, demonstrating that the MAPE is within 0.15% in this range of eigenvalues. The range of MAPEs is lowest for the ground state and shows a slight increasing trend with higher energy levels. This increase can be attributed to the magnitude of the eigenvalues decreasing as the energy level increase. Higher energy states, with their inherently lower magnitudes, offer a smaller margin for absolute error, thus demanding greater accuracy. The model achieves this to a lesser degree, although the level of error is still well within the range of chemical accuracy. The overall MAE averaged over all model systems is 2.22 $\times\,10^{-4}$ Ha and the corresponding averaged MAPE over all eigenvalues is $3.18{\times10^{-2}}\%$ denoted by gray horizontal lines in figures 11 and 12. This underscores the FNO model's ability to accurately capture the potential-density mapping.

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3.2.2. Diatomic molecule with varying charges and bond length

After demonstrating improved performance over the PINNs model for the one-dimensional atom, we investigate the FNO model's abilities for the one-dimensional diatomic molecule. Variations in both charges (Z₁ and Z₂) and bond length d pose a different set of challenges for the FNO model. As depicted in figure 13, the plots for eigenvalue errors indicate a slightly higher margin of error in comparison to the atomic model system. The eigenvalue errors consistently remain within chemical accuracy, except for a particular outlier. Figure 14 shows the MAPE range for each eigenvalue. There is a clear trend of increasing MAPE range from the ground state to excited states. Nevertheless, the overall MAE of 2.27 $\times\,10^{-4}$ Ha and MAPE of $1.90{\times10^{-2}}\%$ (denoted by gray horizontal lines) align with the results of the atomic model system, confirming the FNO model's adaptability and accuracy across this more challenging set of system parameters.

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3.2.3. Extrapolation for diatomic molecule

The true test of a predictive model lies in its ability to generalize beyond the parameters of the systems in the training dataset. Figure 15 showcases the eigenvalue error plots for this dataset, which include systems with charges (Z₁ and Z₂) and bond lengths (d) that are outside the previous datasets. The errors recorded were marginally higher than those for the training dataset, and about half the eigenvalues are predicted within chemical accuracy. The percentage errors are within 2%, as depicted in figure 16. The FNO model yields overall an MAE of 5.4 $\times\,10^{-3}$ Ha and MAPE of $0.25 \%$ (denoted by gray horizontal lines). This demonstrates the FNO model's robustness and potential for broader applicability in solving the KS inversion problem.

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