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iPINN for Broadband CARS Phase Retrieval: A Framework for Function Approximation and Inverse Modeling Problems in Nonlinear Spectroscopy
Ravi Teja Vulchi, Carl Messerschmidt, Mohammadsadegh Vafaeinezhad, Rajendhar Junjuri, Tobias Meyer-Zedler, Juergen Popp, Thomas Bocklitz
TL;DR
BCARS phase retrieval is an ill-posed inverse problem because resonant Raman information is distorted by coherent mixing with a varying non-resonant background. iPINN predicts Lorentzian peak parameters and reconstructs susceptibility through a differentiable physics model, achieving the lowest benchmark error and depth-invariant accuracy in five of seven solvents.
Problem
BCARS spectra are distorted by interference between resonant signals and the non-resonant background, producing asymmetric line shapes that obscure directly interpretable Raman information.
Method
iPINN uses a transformer encoder to predict Lorentzian peak parameters and a differentiable analytical Lorentz forward model to reconstruct resonant susceptibility during end-to-end training.
Results
MAE 0.016 was the lowest among tested baselines, and depth-invariant accuracy was observed in five of seven solvents across 28 zero-shot spectra.
Takeaways & Limitations
Inverse parametric prediction with a differentiable physical decoder supports phase retrieval that remains stable across varying non-resonant-background conditions.
Takeaways & Limitations
The fixed maximum of 24 peaks may be insufficient for biological samples with dense, overlapping vibrational bands.
Abstract
from arXiv · showhide
Phase retrieval in broadband coherent anti-Stokes Raman spectroscopy (BCARS) is an ill-posed inverse problem. The Raman-like signal is encoded in the imaginary part of the resonant susceptibility, which mixes coherently with a non-resonant background (NRB) that varies across acquisitions. We introduce an inverse physics-informed neural network (iPINN) that predicts Lorentzian peak parameters from raw BCARS spectra and reconstructs the resonant susceptibility through a differentiable analytical forward model. A transformer encoder assigns spectral features to 24 learnable peak slots, and a multi-view consistency loss enforces invariance across NRB pattern, NRB strength, and noise. Unlike direct spectral regression approaches, the method retains accuracy under varying acquisition conditions. On a public benchmark, iPINN achieves the lowest error among the tested baselines (MAE 0.016 vs. next-best 0.046). On 28 zero-shot test spectra acquired across seven solvents and four focal positions, accuracy is depth-invariant in five of seven solvents. These results show that inverse parametric prediction with a differentiable physical decoder supports robust phase retrieval across measurement conditions.
Problem Statement and Motivation
BCARS phase retrieval is ill-posed because intensity-only spectra combine resonant Raman information with variable non-resonant background distortions. The paper therefore targets compact, physically grounded peak inference that remains stable across acquisition conditions.
- BCARS spectra contain asymmetric line shapes because resonant signals interfere with the non-resonant background.
- The chemically meaningful Raman-like information resides in the imaginary resonant susceptibility but is not measured directly.
- NRB shape, strength, instrumental broadening, and noise vary across measurements, while molecular structure is represented by a low-dimensional peak set.
- Classical KK and MEM methods commonly require reference NRB spectra or normalization standards and degrade with finite support, non-ideal features, or substantial noise.
- Recent deep-learning methods remain sensitive to synthetic training distributions and are often validated under narrow acquisition conditions.
- iPINN maps measured spectra to amplitudes, centers, and linewidths, then reconstructs the resonant signal through a differentiable Lorentzian model.
Datasets and Methods
The study trains iPINN with synthetic spectra that preserve solvent peak structure while varying measurement distortions, then evaluates the model on public and zero-shot experimental datasets. Its transformer encoder predicts Lorentzian parameters, and physics and consistency objectives constrain the resulting reconstructions.
- Synthetic data generation: Synthetic spectra retain solvent-specific peak templates while resampling NRB profiles, NRB-to-signal ratios, noise, and spectral coverage.
- Forward model: The resonant susceptibility is modeled as a complex sum of Lorentzian modes parameterized by amplitude, wavenumber center, and linewidth.
- Synthetic data generation: Training uses Gaussian, polynomial, and sigmoidal NRB families to reduce dependence on a single background morphology.
- Experimental evaluation: Evaluation includes a five-spectrum public benchmark and a seven-solvent dataset measured across focal positions under variable NRB conditions.
- 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.
- Training objectives: The Lorentz-model loss compares normalized reconstructed imaginary susceptibility against the target, supervising the full spectral signature implied by predicted parameters.
- Training objectives: A multi-view consistency loss requires matching peak predictions across realizations that vary NRB shape, NRB strength, and noise.
Results
Across public benchmark and focus-depth experiments, iPINN reconstructed Raman spectra accurately while remaining comparatively stable under changing SNR, solvent, and NRB conditions. Its physics and consistency losses jointly improved robustness, especially under strong NRB.
- 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.
- Public benchmark: 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.
- Public benchmark: 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.
- 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%.
- 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.
- Ablation study: 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.
Discussion and Future Outlook
iPINN’s parametric, physics-constrained design improves spectral reconstruction and stability under changing NRB conditions, while revealing limitations in forward-model fidelity and peak capacity. Its depth-invariant behavior supports direct comparison of spectra across hyperspectral BCARS images without per-pixel correction.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.