iPINN for Broadband CARS Phase Retrieval: A Framework for Function Approximation and Inverse Modeling Problems in Nonlinear Spectroscopy

• BCARS spectra contain asymmetric line shapes because resonant signals interfere with the non-resonant background.Source: Problem Statement and Motivation · p. 1
• The chemically meaningful Raman-like information resides in the imaginary resonant susceptibility but is not measured directly.Source: Problem Statement and Motivation · pp. 1–2
• NRB shape, strength, instrumental broadening, and noise vary across measurements, while molecular structure is represented by a low-dimensional peak set.Source: Problem Statement and Motivation · p. 2
• Classical KK and MEM methods commonly require reference NRB spectra or normalization standards and degrade with finite support, non-ideal features, or substantial noise.Source: The Challenge of Inverse Modeling in… · p. 2
• Recent deep-learning methods remain sensitive to synthetic training distributions and are often validated under narrow acquisition conditions.Source: The Challenge of Inverse Modeling in… · p. 2
• iPINN maps measured spectra to amplitudes, centers, and linewidths, then reconstructs the resonant signal through a differentiable Lorentzian model.Source: The Inverse PINN Framework (iPINN) · p. 2

Synthetic data generation

• Synthetic spectra retain solvent-specific peak templates while resampling NRB profiles, NRB-to-signal ratios, noise, and spectral coverage.Source: Procedural Data Generation Pipeline · pp. 3, 5

Forward model

• The resonant susceptibility is modeled as a complex sum of Lorentzian modes parameterized by amplitude, wavenumber center, and linewidth.Source: Procedural Data Generation Pipeline · p. 3

Synthetic data generation

• Training uses Gaussian, polynomial, and sigmoidal NRB families to reduce dependence on a single background morphology.Source: Procedural Data Generation Pipeline · pp. 3, 5

Experimental evaluation

• Evaluation includes a five-spectrum public benchmark and a seven-solvent dataset measured across focal positions under variable NRB conditions.Source: Experimental Datasets and Evaluation… · p. 5

Model architecture

• The inverse model maps spectra to peak parameters through a transformer encoder and reconstructs the resonant signal with a differentiable analytical Lorentz model.Source: Figure 1 · p. 4, Procedural Data Generation Pipeline · p. 4, Model Architecture · p. 8

Training objectives

• The Lorentz-model loss compares normalized reconstructed imaginary susceptibility against the target, supervising the full spectral signature implied by predicted parameters.Source: Training Objectives · p. 10
• A multi-view consistency loss requires matching peak predictions across realizations that vary NRB shape, NRB strength, and noise.Source: Training Objectives · p. 10

Public benchmark

• MAE 0.019 for 2C and 0.025 for 3C were achieved on toluene, with major peaks recovered and cleaner baselines than the compared reconstructions.iPINN avoided false positives and residual structure in the silent region, although it mildly underestimated the C–H stretching cluster relative to the 1003 cm−1 ring-breathing mode.Source: Generalization across the BCARS Phase… · pp. 10–11
• iPINN reconstructed DMSO consistently as SNR decreased, while baseline models showed more residual background and spurious weak-region features.The DMSO spectra contained weak resonances over a dominant NRB envelope, making retrieval more difficult.Source: Generalization across the BCARS Phase… · p. 11
• MAE 0.0156 and RMSE 0.0489 were iPINN’s lowest mean errors across five benchmark spectra, with the smallest error-bar spread.VECTOR was next-best at MAE 0.046; SpecNet and GAN followed at 0.047 and 0.058.Source: Generalization across the BCARS Phase… · p. 12

Focus-depth stability

• Five of seven solvents showed near-flat MAE across four focus depths, while toluene and DMSO had weaker depth invariance concentrated at the deepest-focus value.CyHex had CV 2.6%, EtOH 0.6%, toluene 18.6%, and DMSO 34.7%.Source: Stability across NRB Conditions on the… · p. 12

Ablation study

• Removing the physics loss was the dominant failure mode under strong NRB, increasing depth sensitivity relative to the full model.For methanol at 3 µm, no-Both reached MAE 0.0302 versus 0.0163 for no-Cons; on ethanol, full iPINN reached CV 0.6% versus 50.6% for no-Physics.Source: Ablation Study: Contributions of… · pp. 12, 14
• The full iPINN had the lowest CV in five of seven solvents, and both auxiliary losses acted as complementary regularizers under varying NRB strength.For ACN and DMSO, an ablation achieved lower CV despite the full model’s small absolute MAE values.Source: Ablation Study: Contributions of… · pp. 12, 14

Structural advantages

• iPINN reconstructs spectra from predicted peak parameters, suppressing false peaks and residual background structure by construction.The differentiable Lorentzian decoder produces smooth peaks and near-baseline silent regions instead of independently predicting every spectral point.Source: Discussion and Future Outlook · p. 14

NRB handling

• Predicting peak parameters removes the need to explicitly estimate the NRB during training.The physics loss compares the spectrum reconstructed from θ with the measurement, while NRB variation enters through data generation.Source: Discussion and Future Outlook · p. 14

Ablation findings

• 46% higher MAE occurred when the physics loss was removed for methanol at 3 µm, where spurious peaks appeared under strong NRB.The physics loss suppresses false peaks, whereas the consistency loss controls relative amplitudes of closely spaced peaks; for ethanol at 9 µm, no-Physics had 36% lower MAE than no-Both.Source: Discussion and Future Outlook · p. 14

Robustness

• The full iPINN achieved the lowest coefficient of variation in five of seven solvents as NRB changed across focus depth.The depth-MAE curves remained flattest with both auxiliary losses active.Source: Discussion and Future Outlook · p. 16

Limitations

• The full model was best on average but not uniformly across all seven solvents because its Lorentzian forward model cannot fully reproduce every measured NRB envelope.ACN and DMSO had lower CV for an ablation variant, reflecting residual envelope or peak-height mismatches under changing focus.Source: Discussion and Future Outlook · p. 16

Future outlook

• The method is limited by a fixed maximum of 24 peaks, which may be insufficient for biological samples with dense, overlapping vibrational bands.The paper proposes adaptive peak budgets and forward-model refinement as extensions toward chemically heterogeneous and spectrally dense samples.Source: Discussion and Future Outlook · p. 16

Practical implications

• Depth-invariant behavior could make spectra directly comparable across hyperspectral BCARS images without per-pixel correction.The practical setting is pixel-to-pixel variation in effective NRB within an image.Source: Discussion and Future Outlook · p. 16

Future outlook

• Parametric Lorentzian neurons could replace the current linear spectrum tokenizer to encode spectroscopic structure directly at the input.The proposed input-side prior mirrors the differentiable Lorentz model at the output and is intended to reduce dependence on training data for expressing spectroscopic structure.Source: Discussion and Future Outlook · p. 16