Using machine learning to map simulated noisy and laser-limited multidimensional spectra to molecular electronic couplings

Abstract

Two-dimensional electronic spectroscopy (2DES) has enabled significant discoveries in both biological and synthetic energy-transducing systems. Although deriving chemical information from 2DES is a complex task, machine learning (ML) offers exciting opportunities to translate complicated spectroscopic data into physical insight. Recent studies have found that neural networks (NNs) can map simulated multidimensional spectra to molecular-scale properties with high accuracy. However, simulations often do not capture experimental factors that influence real spectra, including noise and suboptimal pulse resonance conditions, bringing into question the experimental utility of NNs trained on simulated data. Here, we show how factors associated with experimental 2D spectral data influence the ability of NNs to map simulated 2DES spectra onto underlying intermolecular electronic couplings. By systematically introducing multisourced noise into a library of 356 000 simulated 2D spectra, we show that noise does not hamper NN performance for spectra exceeding threshold signal-to-noise ratios (SNR) of ca. 12.4, 2.5, and 5.1 if uncorrelated additive, correlated additive, or intensity-dependent noise sources dominate, respectively. In stark contrast to human-based analyses of 2DES data, we find that the NN accuracy improves significantly (ca. 84% → 96%) when the data are constrained by the bandwidth and center frequency of the pump pulses. This result is consistent with the NN learning the optical trends described by Kasha's theory of molecular excitons. Our findings convey positive prospects for adapting simulation-trained NNs to extract molecular properties from inherently imperfect experimental 2DES data. More broadly, we propose that machine-learned perspectives of nonlinear spectroscopic data may produce unique and perhaps counterintuitive guidelines for experimental design.

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1 Introduction

Coherent multidimensional spectroscopies (CMDS) afford rich insight into the mechanisms of light-driven molecular processes.^1–7 For example, studies using two-dimensional electronic spectroscopy (2DES) in the last two decades exposed the central role that electron–vibrational (vibronic) coupling plays in the excited-state photophysics of chemical and material systems, including natural photosynthetic complexes,^8–11 organic semiconductors,^12–16 and quantum dots.^17,18 The abundance of information within 2DES spectra, as with spectra from other CMDS techniques, comes at the expense of interpretability; results of early 2DES measurements sparked decade-long debates of their physical interpretation.^7,19–22 Developing robust methods to derive accurate chemical information from 2DES will be indispensable as this technique is used increasingly to probe complex, device-relevant condensed-phase systems.

Spectroscopy is often used to solve inverse problems, where physical insight about a chemical system is sought from spectroscopic data. Machine learning (ML) models are uniquely suited to solve inverse problems,^23,24 and ML has already been applied to many inverse chemical problems in spectroscopy.^25–40 For example, Lansford et al.²⁷ and Enders et al.²⁸ used ML to extract surface microstructure and functional group information, respectively, from infrared spectra. Cui et al.⁴¹ demonstrated an ML method that relates infrared and Raman spectra to the electrocatalytic properties of CO₂ reduction. Despite recent progress in joint ML-spectroscopic approaches, time-evolving nonlinear spectra are vastly more complicated than steady-state linear spectra, and this is especially true for spectra derived from multidimensional methods like 2DES. As a result, few studies^29,31–36 have demonstrated how ML can be used to map the properties of molecular systems directly from their multidimensional spectra. However, innovations enabled by ML applied to linear spectroscopy^{27,28,38,41,42} and magnetic resonance spectroscopies^37,43,44 clearly indicate the potential of using ML to transform the interpretation of complicated spectroscopic data.

The data requirements of ML pose a significant challenge in applying ML to spectroscopy.^24,29,43,45 There is currently no public repository for experimental 2DES data. Of the experimental data that accompany journal publications, factors such as low molecular diversity, variation in data processing methods, and insufficient sample characterization hinder the prospects for training neural networks (NNs) with purely experimental 2DES datasets. A viable and precedented alternative is to use simulated data to train NNs for experimental applications.^{27,29,46–52} Simulated data offer the unique advantages of practically infinite availability and complete knowledge of the underlying physical properties, which enabled several recent studies^{29,31,33–36} that leverage ML to solve inverse problems with multidimensional spectra. Simulated data are, however, pristine: they do not typically include the influence of experimental features in CMDS spectra, such as noise, finite pulse bandwidths, and imperfect laser-sample resonance conditions.^2,51,53–55 It remains unknown how such experimental aspects of 2DES might influence the performance of ML-based interpretation tools. This gap in knowledge contributes to the already considerable challenge of adapting simulation-trained NNs to experimental applications.

Here, we develop an expansive database of 356 000 vibronic dimer 2DES spectra and use it to identify how experimental constraints influence inverse problem solving with a feed-forward NN. When trained and evaluated on pristine simulated data, the NN classifies unseen spectra to one of 33 electronic coupling categories with ∼84% accuracy. By systematically introducing experimental constraints, or “data pollutants,” into the spectra and performing repeated training and evaluation, we find how the pollutants influence the testing performance of the NN. We find that the simulation-trained NNs are relatively robust to additive noise sources with correlations along the probe axis (e.g., intensity jitter of the local oscillator) and sources that depend on the signal magnitude (e.g., fluctuations in the pump power or beam alignment). The NN performance appears to be most susceptible to uncorrelated additive noise sources, such as detector dark current or the readout electronics. We also find that NN performance increases significantly (up to ∼96% accuracy) when the effects of pump bandwidth and center frequency are accounted for in the spectral dataset. We find that this counterintuitive result provides fundamental insight into the machine learnability of electronic coupling information in multidimensional optical spectra. Ultimately, our study clarifies how the performance of simulation-trained NNs may vary, potentially in a positive or negative direction, as they are adapted for experimental applications. These findings encourage the use of ML to derive chemical insight directly from multidimensional spectroscopy experiments.

2 Methods

2.1 Spectral database for machine learning

We performed nonlinear response simulations in Python to generate our training and testing datasets. Because of computational costs and storage limitations, we limited the scope of the current study to models for molecular dimers. Studies from the last two decades found that simple molecular models, such as the harmonic oscillator or purely electronic dimer models, are often insufficient to describe sub-picosecond photophysics.^8,10,56,57 Hence, we used a Holstein-like vibronic exciton Hamiltonian, which was shown to be accurate for predicting features in experimental 2DES spectra of light-harvesting systems.^{8,12,13,58–60} The system Hamiltonian is

H_sys = H_el + H_vib + H_el−vib, (1)

where H_el and H_vib are the electronic and vibrational Hamiltonians, and H_el–vib describes the electron–vibrational coupling. The electronic portion of eqn (1) for a molecular dimer is written in the Condon approximation aswhere ε_n is the electronic transition energy for molecule n, and c_n are the electronic creation and annihilation operators, respectively, such that represents an exciton on site n, and J_Coul is the coulombic coupling. The vibrational and vibronic Hamiltonians are:where creates (annihilates) vibrational quanta for vibration m with frequency ℏω_m and Huang–Rhys factor λ_m².

In generating the spectral database with the vibronic exciton Hamiltonian (eqn (1)), we set ranges for all Hamiltonian parameters so that the simulated spectra correspond to molecular systems that are typically studied with 2DES. Fig. 1 shows the parameter distributions for the coulombic couplings and the nuclear displacements. We varied the coulombic coupling from J_Coul = −800 to +800 cm⁻¹ (Fig. 1a), which corresponds to strong J- and H-type coupling interactions, respectively.⁶¹ We previously found that NNs disproportionately misclassify the value of J_Coul that underpins the 2DES spectra of J-type dimers.^33,34 Thus, while we primarily used a 50 cm⁻¹ increment as J_Coul was varied, we used smaller increments in varying J_Coul < −550 cm⁻¹ (see Fig. 1a).

Fig. 1 Values of the (a) coulombic coupling (J_Coul) and nuclear displacements (λ_i) of the (b) i = 1300 and (c) i = 200 cm⁻¹ modes (eqn (2) through (4)) used in generating the spectral database. There are 356 000 unique 2DES spectra in the full dataset, reflecting 1424 unique homodimers. Slice areas in each hollowed circle are proportional to the amount of data they represent. Outward-facing ticks in (a) indicate the boundaries of the 33 classes reflected in the output of the neural network (vide infra). See Table S1† for further details.

We made two compromises to balance storage costs with the generality of our data space. First, we considered only homodimers (i.e., ε₁ = ε₂ = ε in eqn (2)). We chose the specific value of ε = 14 500 cm⁻¹ to align with the approximate transition energy of terrylenediimide, a prototypical organic chromophore with extensive prior 2DES characterization.^{13,59,62–64} Second, we constrained eqn (3) and (4) to include two independent vibrational modes. In a previous study, we considered systems with up to three vibrational modes and analyzed how the number and frequency of the modes impacted the machine learning.³³ Here, we prioritize the inclusion of one high-frequency (1300 cm⁻¹) and one low-frequency (200 cm⁻¹) mode. This is because high-frequency modes, especially C=C stretches, often exhibit significant Franck-Condon (FC) activity in organic chromophores.⁶¹ Also, low-frequency modes are found to play significant roles in non-adiabatic excited-state dynamics.^65–68 Further details of the spectral database are provided in the ESI.†

2.2 2DES simulations

We simulated absorptive 2DES signals (e.g., Fig. 3a) for each model Hamiltonian using in-house Python codes (freely available in ref. 69). We calculated the third-order optical response functions (ground-state bleach and stimulated emission pathways) as a function of the t₁, t₂, and t₃ interpulse time delays (Fig. 2a). We applied a phenomenological lineshape function⁷⁰ to each dimension of the time-domain signals to account for phenomenological system–bath interactions and to realize finite linewidths. The final absorptive 2DES spectra are computed by fast Fourier transformation of the signal to the pump (ω₁/(2πc)) and probe (ω₃/(2πc)) frequency domains (abbreviated herein for clarity as ω₁ and ω₃, respectively). Table S2† shows the parameters that were used in our nonlinear response simulations. We selected parameters that reflect common scenarios encountered in 2DES experiments (e.g., spectral linewidths, time and frequency resolutions, etc.). Further details of the simulations are described in the ESI.†

Fig. 2 Schematic workflow of the spectral simulations, data processing, and machine learning trial employed here. (a) We used nonlinear response function simulations to generate a spectral database for all systems within the parameter space portrayed in Fig. 1. (b) For each type of data pollutant, we operated on a copy of the clean spectral database and sent the polluted spectra to the ML algorithm. (c) We used 80% of the data to train a categorical feed-forward neural network and the remaining 20% for testing.

2.3 Data pollution

The simulations described above provide “clean” spectra, which do not capture many features of experimentally measured 2DES spectra. Noise and pulse properties can significantly influence the results of 2DES experiments,^2,53–55 yet such factors are commonly neglected in simulations. To explore: (i) how experimental effects (i.e. noise and laser-sample resonance constraints) influence the machine-learnability of 2D data and (ii) bridge simulation-trained NNs toward applications to experimental data, we “polluted” our ML datasets prior to both training and testing and examined the resulting effects on NN performance. Fig. 2b shows the strategy for introducing each kind (vide infra) of pollutant; we applied the pollution operation to a copy of the pristine dataset, trained the ML model on the polluted data (Fig. 2c), and then computed the performance on a test set of the polluted data.

Noise signatures, and the spectral characteristics of the pump pulses, are key factors that augment experimental 2DES spectra compared to their simulated counterparts. In nonlinear spectroscopy experiments, noise manifests in numerous ways that can vary depending on the signal acquisition geometry and any procedures used for background removal (e.g., chopping and phase cycling).^{51,55,71–73} Noise signatures are commonly categorized as either “additive” or “multiplicative” in nature (Fig. 3b). Additive noise refers to signal-independent fluctuations arising from sources such as local-oscillator intensity jitter, detector dark current, and readout electronics.^55,71,74 In contrast, multiplicative or “convolutional” noise sources are proportional to the analyte signal (∝ χ⁽³⁾).^71,72 Examples of multiplicative noise, which we will denote as “intensity-dependent” noise herein, are shot noise in the pump pulses and fluctuations in the beam overlap at the sample.⁷³

Fig. 3 (a) A representative “clean” spectrum generated with the parameters provided in the inset table. We polluted the datasets by (b) adding one of three types of experimental noise or (c) convoluting the 2DES signal with a Gaussian pump pulse. Representative images of the isolated data pollutants are shown in the upper panels of (b) and (c); the lower panels of (b) and (c) show the resulting polluted spectra. All spectra and noise profiles are plotted against the color scale in (a).

Additive noise typically dominates the total noise present in a 2DES experiment, and it is instructive to distinguish between two unique scenarios. In most set-ups, intensity jitter in the local oscillator pulses leads to constant baseline offsets along ω₃ that fluctuate along ω₁ (e.g., the left-most panel of Fig. 3b).^51,71–73 We refer to this class of noise as correlated additive noise herein. In contrast, detector dark current and read-out electronics contribute noise that is uncorrelated between pixels, or Gaussian pixel noise (e.g., the middle panel of Fig. 3b).

For each unique system Hamiltonian, we modeled noise at every ω₁ × ω₃ × t₂ data point using a normal distribution centered around zero and with a standard deviation of σ. Unlike intensity-dependent noise and uncorrelated additive noise, which are normally distributed along all dimensions, we constructed correlated additive noise profiles that are Gaussian distributed only along ω₁ × t₂. All 2DES spectra associated with a given model Hamiltonian were normalized to the maximum signal magnitude at t₂ = 0 prior to noise injection. As such, a value of σ = 1 corresponds to random noise comparable to the signal magnitude (or SNR ≈ 1 at t₂ = 0). Table 1 provides the values of σ that we considered in this study, divided into additive (σ_add) and intensity-dependent (σ_int) categories. Both correlated and uncorrelated additive noise profiles are simply added to the 2D spectral data. In contrast, for intensity-dependent noise, we multiply each 2D noise profile (size n_ω1·n_ω3, where n_ω1 and n_ω3 are the number of “pixels” in the pump and probe frequency dimensions, respectively) element-wise by the corresponding 2D spectrum prior to addition. Note that for a given value of σ_add, the SNR of the noised spectra may differ slightly between correlated and uncorrelated additive noise (Fig. S3†). See the ESI† for details of the noise injection procedures.

Table 1 Variables and values therein for each form of data pollutant

Data pollutant	Parameter (units)	Values
Additive noise	σ add (unitless)	0, 1 × 10^−5, 2.5 × 10^−5
		5 × 10^−5, 7.5 × 10^−5
		1 × 10^−4, 2.5 × 10^−4
		5 × 10^−4, 7.5 × 10^−4
		0.001, 0.0025, 0.005
		0.0075, 0.01, 0.025
		0.05, 0.075, 0.1, 0.25
Intensity-dependent noise	σ int (unitless)	0, 0.001, 0.0025, 0.005
		0.0075, 0.01, 0.025
		0.05, 0.075, 0.1, 0.25
		0.5, 0.75, 1, 2.5, 5
		7.5, 10, 25, 50
Pump spectrum	Δω (cm^−1)	100, 250, 500, 1000, 1500
		2000, 2500, 3000, 3500
		4000, 5000, 7500, 10 000
	ω c (cm^−1)	12 000, 12 250, 12 500
		12 750, 13 000, 13 250
		13 500, 13 750, 14 000
		14 250, 14 500, 14 750
		15 000, 15 250, 15 500
		15 750, 16 000, 16 250
		16 500, 16 750, 17 000

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2DES signals depend critically on the spectral overlap between the pump pulses and the sample absorption. Both the spectral bandwidth (Δω) and center frequency (ω_c) of the pump pulses determine the spectral overlap. To introduce pump pulse characteristics to our ML dataset, we convoluted the simulated 2DES spectra with Gaussian pulses (eqn (S9)†) parameterized with realistic values of ω_c and Δω (Table 1). We defined the former to span the excited-state transition energies of the molecular systems represented in our spectral database (ca. 12 000 to 18 500 cm⁻¹). Depending on the experimental apparatus, the Δω of the pump pulses in conventional 2DES experiments typically ranges between 1000 and 6000 cm⁻¹.^64,75,76 See the ESI† for further information.

2.4 Machine learning

The machine-learning protocols used here are based on earlier workflows of Parker and coworkers³³ that use the PyTorch library⁷⁷ in Python. Our codes are freely available to the public in ref. 78. Here, we examined an inverse problem where we trained feed-forward NNs (Fig. 2c) to classify 2DES spectra based on the electronic couplings in the underlying model Hamiltonians. The NN uses flattened 2DES spectra (1D arrays of length n_ω1·n_ω3) as inputs. We used an automated trimming algorithm on all spectra (see the ESI† for details) to remove outer low-intensity signals and to ensure that all final spectra (i.e. NN inputs) have size: n_ω1 = n_ω3 = 151. The NN applies linear transformations and rectified linear unit (ReLU) activation functions to connect the input layer (consisting of 22 801 neurons from the spectra of size 151 × 151) to a single hidden layer with 300 neurons. Additional hidden layers produced marginal performance gains, as discussed in the ESI.† We apply a dropout operation on hidden layer neurons for regularization. Linear transformations and softmax activation functions connect the hidden layer and output layer neurons. Each of the 33 neurons in the output layer corresponds to a single class of electronic coupling J_Coul (see Fig. 1a for the class bounds in the J_Coul parameter space).

We conducted independent ML trials for each polluted dataset (i.e., the NN was trained and tested on each polluted dataset). For simplicity, we determined a set of hyperparameters (Table 2) that optimizes NN performance when trained and tested on clean data. We then kept the hyperparameters constant for all ML trials with the polluted datasets. We also used the same initializations for the trainable parameters (e.g., weights and biases) in each trial and kept the training and testing subsets consistent by seeding data shuffling and splitting operations. See the ESI† for additional details of our ML procedures and hyperparameter optimization (Fig. S2†).

Table 2 Hyperparameters used for all NN trials in this study

Hyperparameter	Value
a Optimized with a grid search for the unpolluted dataset (see Table S3 and Fig. S2).
Activation function	ReLU
Training-testing split	80 : 20
Learning rate^a	0.001
Number of hidden layers	1
Hidden layer size^a	300
Epochs	30
Dropout probability^a	0.2
Batch size	100

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In addition to the conventional accuracy metric that assesses NN performance, we used the scikit-learn module⁷⁹ in Python to calculate F1 scores and top-k accuracies. Compared to accuracy, the F1 score provides better accounting of false positives and false negatives, as well as more robustness to class imbalances.⁸⁰ The top-k accuracy examines whether the true classification is in the top k most probable classifications predicted by the NN. Thus, the top-k accuracy provides additional insight into the precision of NN classifications (e.g., how far the misclassifications are from ground truth).

3 Results and discussion

The quality of NN classifications when trained and tested on clean (not polluted) spectra is a key reference point for this study. We found that the NN classifies clean 2DES spectra in their correct J_Coul category with an accuracy of 83.99% and an F1 score (macro-averaged) of 0.845. This high performance is consistent with our previous study,³³ in which we found an accuracy of ca. 92% for a similar J_Coul range subdivided into five categories (as opposed to the 33 used here). Note that, in general, we observed that the accuracy and F1 scores were approximately equal (within about one percentage point, as shown in Fig. S8†). For clarity, we only report the F1 scores.

Fig. 4 shows the performance of the NN trained and tested using clean spectra through the lens of a confusion matrix. In the confusion matrix representation, correct and incorrect NN classifications are reflected by on- and off-diagonal values, respectively. We observe that while 16% of the NN classifications are incorrect, the majority of misclassifications occur only one category away from the ground truth. This observation is consistent with the calculated 99.04% top-2 accuracy.

Fig. 4 Confusion matrix comparing the true vs. NN-predicted values of J_Coul when trained and tested on clean data. Each row is normalized to unity. Diagonal entries, indicated by the dotted white line, reflect correct classifications; off-diagonal entries report on misclassifications.

3.1 Influence of noise on NN performance

The dependence of the NN performance on the amount of additive noise in the dataset is shown in Fig. 5a (see the insets for representative 2DES spectra). We find that training and testing F1 scores are relatively unaffected (remaining within 5% of the F1 ≈ 0.845 observed on the clean dataset) by both types of additive noise until σ_add exceeds a threshold value, τ_add. The threshold level of uncorrelated additive noise, or τ_add,uc, is 0.0005 (corresponding to SNR ≈ 12.4). In contrast, the threshold for correlated additive noise (τ_add,c) is 0.0025, or SNR ≈ 2.5. This indicates that, with respect to additive noise, the NN performance is more robust to correlated sources than it is to uncorrelated sources. This is further supported by the testing F1 scores at increasingly large values of σ_add, which drop exponentially for datasets with uncorrelated additive noise. The testing scores also decrease with increasing σ_add for models trained on data with correlated noise, but at a significantly slower rate.

Fig. 5 Performance of NNs trained and tested on datasets with varying amounts of (a) additive and (b) intensity-dependent noise. Representative 2DES spectra are included as insets with arrows pointing to the dataset from which they are derived.

Comparing the models' F1 scores on the training versus the testing datasets provides information about over-fitting.⁸¹ For both correlated and uncorrelated sources, we observe small amounts of over-fitting to the training data when σ_add is less than the respective τ_add, as indicated by the slightly higher training F1 scores compared to the testing scores (Fig. 5a). However, as σ_add increases beyond the respective τ_add, the extent of over-fitting substantially increases solely in the case of uncorrelated additive noise. This result suggests that the NNs ‘memorize’ the uncorrelated noise signatures in the training dataset. In ML approaches applied to other types of datasets, noise injection is commonly performed^82–85 to improve the generalizability of NNs (e.g., to mitigate over-fitting⁸¹). Still, previous studies^86–88 found that deep neural networks (DNNs) tend to over-fit when trained on data with noisy labels. This tendency was shown to evince a shift in the DNN from learning general features of the training data to memorizing the noise patterns.⁸⁸ The trends in Fig. 5a, as well as Fig. S7,† suggest a similar effect when the feed-forward NN here is trained on spectra with uncorrelated additive noise. Still, even for σ_add = 0.25, the NN significantly outperforms random guessing.

As in the case of additive noise, we find that the training and testing F1 scores are invariant with increasing intensity-dependent noise (Fig. 5b) until a threshold is exceeded, i.e. σ_int > τ_int = 0.25 (corresponding to SNR ≈ 5.1). The intensity-dependent threshold is thus significantly higher than the thresholds for both types of additive noise (i.e. τ_int > τ_add,c > τ_add,uc). This makes sense, as increasing σ_add leads to an increase in SNR more quickly than increasing σ_int (see Fig. S3†). With respect to SNR, the impact of intensity-dependent noise on NN performance falls between that of correlated and uncorrelated additive noise (SNR ≈ 5.1 vs. 2.5 and 12.4, respectively). Fig. 5b also shows that, for σ_int > τ_int, the NN performance exhibits a logistic-like decay with increasing intensity-dependent noise (compared to the exponential decay found for additive noise). In contrast, the training F1 score shows a slight growth from 0.882 to 0.913 between σ_int = 0.5 and 5, respectively, followed by an exponential decay for σ_int > 5. Aside from indicating over-fitting, this result suggests fundamental differences between the nature of over-fitting for spectral datasets with uncorrelated additive vs. intensity-dependent noise.

The results in Fig. 5 show that each category of noise explored here exhibits clear and distinct influences on NN classifications of 2D spectra based on electronic couplings. We find that the NN performance is generally robust up to certain threshold amounts of noise (τ_add,uc = 0.0005, τ_add,c = 0.0025, and τ_int = 0.25). These thresholds predict reductions in NN performance for spectra with SNR < 12.4, 2.5, and 5.1 if the total noise is dominated by uncorrelated additive, correlated additive, or intensity-dependent noise, respectively. For σ > τ, the NNs generally exhibit a mixture of learning and memorizing, with memorization being particularly evident for datasets with uncorrelated additive or intensity-dependent noise. Many of the misclassifications when σ > τ, as shown in Fig. S9,† occur more than one category away from the true class. This is especially the case for spectra from Hamiltonians that have weak-to-intermediate electronic coupling values.

3.2 Influence of pump characteristics on NN performance

Resonance between the pump pulses and the absorption spectrum of the sample in a 2DES experiment critically determines the magnitude and shape of features in the 2DES spectra. As described above and shown in Fig. 3c, we varied the spectral bandwidth (Δω) and center frequency (ω_c) of the pump pulses to simulate experiments with varied resonance conditions. The heatmap in Fig. 6a shows the testing F1 score after training and testing the NNs with datasets spanning each combination of the Δω and ω_c parameters. We observe rich variation in the NN performance as Δω and ω_c are varied. For all ω_c, the F1 scores when Δω = 10 000 cm⁻¹ are similar to those obtained from the clean dataset (ca. 0.845). As Δω decreases, we observe that the F1 scores increase and subsequently decrease. The values of Δω that yield the maximum F1 score depend strongly on ω_c. Several combinations of Δω and ω_c yield F1 scores above 0.95 (dark red regions). Within the range 500 ≤ Δω ≤ 5000 cm⁻¹, the F1 scores are bi-modal with respect to ω_c. For ω_c ≤ 14 000 cm⁻¹ and ≥15 000 cm⁻¹, small values of Δω result in F1 scores below the 0.845 score obtained with the clean dataset (blue regions). All trends noted in Fig. 6a are also found in the training F1 scores (Fig. S10†).

Fig. 6 (a) NN F1 score for the testing data as a function of Δω and ω_c of the pump pulses. The color scale is relative to the F1 score of 0.845 found from the clean dataset (red and blue indicate higher and lower F1, respectively). The upper panel illustrates the expected optical responses of purely electronic J- versus H-type aggregates in Kasha's exciton model. The dashed line indicates exact resonance between the center-frequency of the pump pulses and the monomer optical response. (b) Example 2DES spectra and F1 scores from the corresponding datasets for the (Δω, ω_c) coordinates of the matching shape in (a).

The dependencies of the F1 score on Δω and ω_c are counter-intuitive for two reasons. First, 2DES experiments are typically designed with maximal pulse bandwidth.^53,75 This is because lower values of Δω constrain the shape of the 2DES signal along the pump axis, in turn obscuring information about the molecular system. For example, compare the upper and middle spectra in Fig. 6b to 3a. In contrast, we find that smaller Δω values improve NN performance (to a limit). Second, we might expect better NN performance when the pump pulse spectra are resonant with the excited-state energy of the monomers in the site basis (i.e., ω_c = ε = 14 500 cm⁻¹ in eqn (2)). Instead, we find that, for almost all Δω values, the F1 scores increase when the pump spectra are significantly red- or blue-shifted away from the monomer transition energy.

Kasha's theory^61,89,90 for the optical responses of molecular aggregates predicts two exciton classifications based on the sign of the coulombic coupling. The theory predicts that the absorption spectrum of a dimer with J_Coul < 0 (J-type) will be red-shifted compared to that of the isolated monomer (illustrated in the upper portion of Fig. 6a). In contrast, dimers with J_Coul > 0 (H-type) yield blue-shifted absorption spectra. The qualitative predictions of Kasha's theory correlate well with both (i) the bimodal dependence of NN accuracy on ω_c and (ii) the symmetry of the bimodal trend about ω_c = ε = 14 500 cm⁻¹. Such a correlation makes sense since, for all pump spectra except those with ω_c = 14 500 cm⁻¹, the pump biases the spectral dataset toward one exciton response regime and, in turn, influences how the NN learns about the underlying electronic couplings. From the trends in the F1 score as Δω is varied, we posit that, for sufficiently large Δω, biasing one exciton regime over the other boosts NN performance by emphasizing the differences in the 2DES signatures of H- vs. J-type aggregates. However, as Δω is decreased, the performance gains from biasing one exciton regime should eventually be overcome by the erasure of information contained in off-resonant regions of the 2DES spectra. We observe this behavior for all ω_c, as the NN performance drops substantially when Δω drops below threshold values (e.g., for Δω < 1500 cm⁻¹ when ω_c = 12 250 cm⁻¹).

The findings of Fig. 6 show that feed-forward NNs more accurately map 2DES spectra to electronic couplings when the datasets are spectrally constrained (polluted by pump resonance). This result marks a significant departure from human-based designs and analyses of 2DES experiments. With few exceptions,⁹¹ spectrally broadband and on-resonance pump pulses are desired for 2DES experiments. Heisler and coworkers⁵⁴ showed that limited resonance between the pump pulses in a 2DES experiment and the absorption spectrum of the molecular monomer can artificially manifest signatures of electronic coherences in the spectra, which are physically impossible for monomeric samples. While such unphysical information may mislead human analysis of 2DES data, our findings show that some constraints on spectral resonance can positively influence the ability of NNs to learn about spectral signatures of electronic coupling.

3.3 Implications for applications to 2DES experiments

ML presents revolutionary opportunities for decoding information from optical data.^25,26 The results of our study suggest that, despite the signal complexity of nonlinear multidimensional spectroscopy, simple ML approaches like the feed-forward NNs can learn information about the underlying molecular properties in the face of experimental realities (noise and pulse resonance conditions). Although each category of noise investigated here degrades NN performance, this is only the case for noise widths that exceed some threshold (σ > τ). For additive noise, the thresholds vary significantly depending on whether the source adds correlated or uncorrelated noise in the ω₁ × ω₃ dataspace. For uncorrelated additive sources, noise levels exceeding σ_add > τ_add,uc = 0.0005, or SNR <12.4, yield NN performance loss. This is compared to the case of correlated additive sources, where τ_add,c = 0.0025 (SNR ≈ 2.5). For noise sources that scale with the intensity of the analyte signal, we find NN performance is unaffected until σ_int > τ_int = 0.5 (SNR < 5.1). We infer from these results that sources of correlated additive and intensity-dependent noise pose limited risk of obscuring coupling information in experimental 2DES spectra.

In practice, the relative influence of different noise sources in CMDS experiments depends critically on factors such as the experimental geometry, detection mechanism, and approaches for background removal. Still, it is generally the case that intensity-dependent noise sources are least problematic.^71–73 In contrast, power fluctuations of the local oscillator, modeled here as correlated additive noise, are often the dominant noise source.⁵¹ In this work, we find that NN performance is less susceptible to correlated additive noise in the training data compared to uncorrelated noise (Fig. 5a). We thus anticipate that noise sources such as the detector (i.e., dark current) and read-out electronics, which contribute uncorrelated additive noise, may be most disruptive to NN-based analyses of CMDS data. Taken together, our findings suggest that NN-based approaches to extracting couplings from experimental data should be robust to noise if the SNR is sufficiently high. Increased reliance on noise-reduction methods, including standard averaging and phase cycling procedures⁹² or post-processing algorithms,⁹³ may be warranted in experimental scenarios prone to excessive noise.

The counterintuitive behavior revealed in Fig. 6 highlights that NNs interface with spectroscopic signals in a fundamentally different way compared to humans. We hypothesize that ML tools may provide opportunities to leverage subtle properties of multidimensional spectra that are overlooked by traditional interpretation methods. For example, the traditional workflow to interpret 2DES spectra for complex molecular systems follows insights gained from nonlinear optical response theories.^1,94,95 Theoretical models predict that cross-peaks in rephasing 2DES spectra are particularly sensitive to electronic and vibronic couplings.^11,56,58,96 In turn, cross-peaks are of central focus in the analysis of experimental 2DES data.^11,97–99 The salient trends in the predictions from nonlinear optical response theories tend to guide human-based analyses of spectra, but there may be a wealth of information contained in the more fleeting trends in the theoretical spectra. Our observation that the NN-interpretability of 2DES data is maximized by sub-optimal (by human standards) resonance conditions supports our hypothesis.

NNs elicit an information-centric perspective of spectroscopic signals during training. In a recent study of Flores and coworkers,³⁸ the authors trained a CNN to classify linear infrared spectra based on functional group information. In addition to spectral features from fundamental vibrational frequencies, they found that the model uses non-intuitive features, such as the absence of specific peaks or peaks from anharmonic modes, in its classifications. Such findings emphasize the potential usefulness of traditionally overlooked properties of spectra in enabling accurate spectral interpretations. Our findings prompt further explorations of how property-specific information is distributed throughout 2DES datasets. Indeed, a recent study of Jakobsson and coworkers¹⁰⁰ found patterns of Fisher information distribution in simulated 2DES spectra that differ from the typical spectral regions that nonlinear response theories suggest for analysis.^11,56,58,96 Information-based (machine-learned in our case) approaches may guide experimental designs or spectral analyses that most efficiently lead to molecular insight from multidimensional spectra.

4 Conclusions

2DES spectroscopy is an increasingly accessible and powerful tool that can probe ultrafast dynamics. Chemically meaningful information is traditionally inferred from 2DES spectra through extensive signal analysis, theoretical modeling, and human-led comparisons of simulated and experimental spectra.^58,98,99 Despite the time and effort required to perform such tasks, misinterpretations of 2DES spectra are still possible and are historically precedented.^7,19–22 Misinterpretations pose a concern, especially as 2DES is used to study increasingly complicated condensed-phase systems. Being agnostic to traditional strategies for interpreting spectra, ML offers a promising route to translate experimental spectra to chemical insight in a robust and data-driven manner. Indeed, there are few studies²⁹ that use ML as an inverse problem solving tool to address experimental 2DES data.

We have shown that even when practical limitations such as noise and pulse resonance conditions are included in the spectral data, feed-forward NNs match simulated 2DES spectra to electronic coupling strengths with high accuracy. We found that uncorrelated additive (e.g., detector dark noise), correlated additive (e.g., intensity fluctuations in the local oscillator), and intensity-dependent (e.g., laser power fluctuations) noise signatures degrade NN performance after threshold amounts of noise are exceeded. The threshold for the latter two categories of noise is significantly higher than for uncorrelated additive noise, suggesting that correlated additive and intensity-dependent noise sources pose a smaller risk of obscuring information about electronic couplings in experimental 2DES spectra. We also found that uncorrelated additive and intensity-dependent noise lead to substantial over-fitting, which aligns with findings of earlier studies of noise with deep neural networks.^85–88 Our results suggest that methods to mitigate the presence of uncorrelated additive noise in 2DES experiments may be necessary to enable ML-driven extractions of molecular properties from measured spectra.

The results presented here convey positive prospects for adapting ML-based tools to analyze and interpret complex experimental 2DES data. Future directions toward ML-guided analyses of experimental spectra may combine polluted simulated data with established transfer learning techniques.^{31,35,36,50,101,102} A potential approach could start with pretraining on polluted simulated spectra to produce a general ML model. Other research groups could then retrain the final layers of the general model (i.e., fine-tune)^50,101,102 with local, smaller experimental datasets. This fine-tuning would allow the model to adapt to the molecular diversity, postprocessing techniques, and noise sources represented in the experimental dataset of interest. Transfer learning techniques have shown promising results in other multidimensional spectroscopy studies focused on protein structure classification.^31,35,36

Finally, this study reveals significant differences between the human- and machine-based interpretation of 2DES signals. In contrast to human-based analysis, we found that NNs exhibit enhanced performance (exceeding an F1 score of 0.96) when the data are constrained by the bandwidth and center-frequency of the pump. We attribute such counterintuitive behavior to the pulse resonance changing how the NN learns the optical properties of molecular excitons. In other words, biasing the spectral data in either of the exciton absorption regimes (J- or H-type) helps the NN learn how couplings manifest in the spectra. This observation provides evidence that NNs accrue a radically different, more information-centric perspective of electronic coupling signatures in 2DES spectra. Further studies of the machine learnability of CMDS spectra may afford guidelines for experimental design as well as approaches to interpret experimental datasets.

Data availability

The codes used for this work are freely available at two public repositories. The machine learning code, iterative data pollution workflow, and a subset of the training/testing dataset are available at https://doi.org/10.5281/zenodo.15041004.⁷⁸ The spectral simulation code that we used to generate the machine learning dataset is available at https://doi.org/10.5281/zenodo.6757663.⁶⁹

Author contributions

The authors confirm contributions to this work as follows: study conceptualization: JDS, KAP; simulation software: JDS, KAP; machine learning infrastructure: JDS, KAP, BS; analysis and interpretation: JDS, KAP, BS; resources: DNB; manuscript preparation: all authors; project administration: JDS, KAP, DNB; All authors approve the final version of the manuscript.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

J. D. S. gratefully acknowledges support from an Arnold O. Beckman Postdoctoral Fellowship in the Chemical Sciences (https://doi.org/10.13039/100000997). Support to K. A. P., B. S., and D. N. B. from the Department of Energy (DE-SC0019400) is acknowledged gratefully.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5dd00125k

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