naman-00/LNO-Computational-Imaging

C-LNO — Lindblad Neural Operator for Inverse Dynamical Imaging

C-LNO is a Lindblad Neural Operator (LNO) developed for inverse dynamical imaging of stochastic Poisson–Nernst–Planck (PNP) ion transport in neuronal systems.

The model combines a Fourier neural-operator backbone with an explicit Lindblad-style dissipative generator and a physical-state projection step. The representation evolves a spatially distributed normalized internal-information structure (R), together with an information-state amplitude/scale.

The model was developed and evaluated as a physics-informed scientific machine-learning method for stochastic dynamical systems and inverse reconstruction.

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Model Summary

Model: C-LNO (R-focused + soft-physics objective)

Architecture: FinalRepairedLNO

Parameters: 546,039

Input spatial resolution: 128 spatial points

Internal state dimension: 6 × 6 matrix representation

Lindblad channels: 4

Fourier modes: 16

Hidden width: 64

Fourier/operator depth: 4

Framework: PyTorch

License: MIT

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Scientific Problem

The underlying task is the reconstruction of dynamical information associated with stochastic ion transport described by a Poisson–Nernst–Planck (PNP) formulation.

The model receives a spatial state representation together with environmental information and predicts the next dynamical state. The objective is not only pointwise reconstruction accuracy, but also preservation of physically meaningful matrix structure during repeated forward evolution.

The internal matrix state (R(x,t)) is treated as a normalized structural representation, while the accompanying amplitude provides an additional information-state scale.

The overall objective is therefore to learn a dynamical operator that is simultaneously useful for inverse reconstruction and robust under repeated application.

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Architecture

C-LNO extends a Fourier neural operator with an explicit Lindblad-style dynamical component.

The forward path consists of:

1. Input projection of the matrix state and environmental variables.
2. Four spectral operator layers combined with pointwise convolutions.
3. A neural generator (G_N) for learned state evolution.
4. A Lindblad generator (G_L) constructed from learned jump operators.
5. Positive learned coupling coefficients for the Lindblad and neural contributions.
6. A physical-state projection that symmetrizes the matrix, enforces positive semidefiniteness, and normalizes the trace.
7. A direct amplitude prediction head.

The environmental input consists of the amplitude, dissipation parameter (\gamma), and stochastic forcing parameter (\sigma). This matches the validated inference architecture used in the project.

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Dynamical Update

The learned state evolution combines neural and Lindblad-style contributions:

$$\mathrm{\Delta }R=\mathrm{\Delta }{R}_{\mathrm{L}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{b}\mathrm{l}\mathrm{a}\mathrm{d}}+\mathrm{\Delta }{R}_{\mathrm{n}\mathrm{e}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{l}}\backslash Delta\; R\; =\; \backslash Delta\; R\_\{\backslash mathrm\{Lindblad\}\}\; +\; \backslash Delta\; R\_\{\backslash mathrm\{neural\}\}$$

with the two contributions scaled by positive learned couplings.

The Lindblad contribution is generated from learned jump operators using the standard dissipative operator structure

$$G_L(R)=\sum _k(L_{k}R{L}_k^T-\frac{1}{2}[L_k^T{L}_{k}R+R{L}_k^T{L}_k])\mathrm{.}G\_L(R)\; =\; \backslash sum\_k\; \backslash left(\; L\_k\; R\; L\_k^\{T\}\; -\; \backslash frac\{1\}\{2\}\; \backslash left[\; L\_k^\{T\}L\_k\; R\; +\; R\; L\_k^\{T\}L\_k\; \backslash right]\; \backslash right).$$

The resulting matrix is passed through the physical-state projection before becoming the next state.

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Physical-State Projection

A central feature of C-LNO is the explicit projection of the predicted matrix state.

The projection:

• symmetrizes the matrix,
• computes its eigendecomposition,
• clips eigenvalues to a positive lower bound,
• normalizes the trace,
• and symmetrizes again.

This enforces the intended positive-semidefinite, unit-trace structure numerically during forward evolution. The validated implementation performs the eigendecomposition in float64 with numerical jitter for robustness.

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Dataset

The final expanded dataset contains:

• 100 independent trajectories
• 5 dynamical/environmental regimes
• 20 trajectories per regime
• 100 time points per trajectory
• 9,900 transition samples
• 128 spatial points
• 6-dimensional internal state representation

The final trajectory-level split contains:

• 70 trajectories for training
• 15 trajectories for validation
• 15 trajectories for testing

corresponding to:

• 6,930 training transitions
• 1,485 validation transitions
• 1,485 test transitions

The split is trajectory-level, so trajectories are not shared between train, validation, and test sets. This avoids temporal leakage between splits.

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Environmental Regimes

The five regimes used in the final evaluation are:

Regime	(\gamma)	(\sigma)
Low noise	0.01	0.02
Stochastic	0.05	0.45
Heavy dissipation	0.35	0.15
Collapse	0.50	0.65
Metastable	0.15	0.30

These regimes are included in both the held-out test evaluation and the long-horizon stability analysis.

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Evaluation

C-LNO was independently evaluated against:

• FNO
• Original LNO
• C-LNO

The comparison uses the same held-out test set.

One-Step Reconstruction

Model	R RMSE	R Relative-L2	Amplitude RMSE	Amplitude Relative-L2
FNO	0.086081	0.516526	0.414353	0.176076
Original LNO	0.086924	0.521581	0.413496	0.175712
C-LNO	0.084614	0.507714	0.418399	0.177796

C-LNO gives the best R reconstruction accuracy among the three models in the final validated comparison.

Relative to FNO, C-LNO reduces R Relative-L2 error by approximately 1.71%.

The amplitude objective shows a small trade-off: the original LNO remains slightly better on amplitude Relative-L2, while C-LNO is optimized toward improved R reconstruction.

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Regime-Wise R Reconstruction

C-LNO achieves lower R Relative-L2 error than FNO in all five evaluated regimes:

Regime	FNO R Relative-L2	C-LNO R Relative-L2
Low noise	0.155526	0.100865
Stochastic	0.622281	0.615386
Heavy dissipation	0.414730	0.403560
Collapse	0.663487	0.659124
Metastable	0.556915	0.549986

Thus, the R-focused C-LNO objective improves the reconstruction of the matrix state consistently across all five regimes.

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Structural Preservation

Unlike a purely accuracy-oriented comparison, this project also evaluates whether the predicted matrix remains physically structured.

The structural diagnostics include:

• trace preservation,
• minimum eigenvalue,
• negative-eigenvalue fraction,
• numerical finiteness.

For the final one-step evaluation, C-LNO preserves the intended matrix structure to numerical precision, with mean trace error on the order of (10^{-8}) and minimum eigenvalues remaining close to numerical zero.

The corresponding FNO predictions can exhibit substantial structural violations.

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Long-Horizon Stability

C-LNO was evaluated under repeated forward application for 100 rollout steps from representative held-out states.

The stability study used:

• 25 representative states
• 5 states per regime
• 100 repeated steps per state
• all five environmental regimes

C-LNO completed the full long-horizon evaluation with finite outputs and preserved the positive-semidefinite/unit-trace structure throughout the tested rollouts.

The long-horizon analysis is a structural stability diagnostic, not a new ground-truth prediction benchmark. The corresponding evaluation protocol explicitly treats it as such.

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Collapse / OOD Stress Testing

A separate stress-test analysis probes the behavior of FNO and LNO under increasingly strong environmental conditions.

The collapse baseline uses:

$$\gamma =0.50,\sigma =0.65\backslash gamma=0.50,\backslash qquad\; \backslash sigma=0.65$$

and additional counterfactual stress levels increase both dissipation and stochastic forcing. These probes are applied to the same collapse input states and are not new ground-truth simulations.

The associated diagnostics include:

• response from baseline,
• maximum trace error,
• minimum eigenvalue,
• negative-eigenvalue fraction.

These measurements should be interpreted as structural stress diagnostics rather than as additional accuracy measurements.

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Training Objective

C-LNO was selected through an R-focused ablation study.

The evaluated objectives included:

1. Baseline reconstruction loss.
2. R-focused reconstruction loss with an additional relative-L2 component.
3. R-focused reconstruction loss with a soft Lindblad-physics regularization term.

The selected C-LNO configuration achieved the strongest validated R reconstruction performance while retaining the structural-preservation behavior of the repaired LNO architecture.

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Intended Use

This model is intended for research in:

• scientific machine learning,
• neural operators,
• inverse dynamical problems,
• stochastic dynamical systems,
• physics-informed machine learning,
• computational imaging,
• PNP-based ion transport modeling,
• operator learning for structured state evolution.

It is particularly intended as a research artifact for studying how physically constrained operator architectures can improve structural stability in learned dynamical systems.

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Limitations

This model should be interpreted within the scope of the experiments reported in the accompanying repository.

The current evaluation is based on simulated stochastic PNP dynamics with a one-dimensional spatial discretization. It does not establish performance on experimental neuronal imaging data or clinical imaging data.

The collapse/OOD analysis is a diagnostic stress test and should not be interpreted as a new supervised accuracy benchmark.

The model also involves an explicit physical-state projection, so structural preservation is partly enforced by the architecture rather than emerging solely from unconstrained learned dynamics.

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Model Files

The Hugging Face repository is intended to provide the exported C-LNO model artifact together with its configuration and documentation.

The model architecture is:

FinalRepairedLNO(
    dim=6,
    width=64,
    modes=16,
    depth=4,
    lindblad_channels=4
)

The exported model contains the learned parameters of the validated C-LNO checkpoint.

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Reproducibility

For reproducibility, the accompanying GitHub repository contains the model implementation, dataset-generation and loading code, experimental results, validation artifacts, and publication figures.

The validated inference architecture and model checkpoint use the same repaired LNO forward path documented in the repository.

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Related Repository

Source code, experiments, validation results, publication figures, and supporting artifacts:

GitHub: https://github.com/naman-dixit00/Lindblad-Neural-Operators-for-Inverse-Dynamical-Imaging-of-Stochastic-Poisson-Ne

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Dataset

The underlying stochastic PNP dataset used in this project is associated with the Harvard Dataverse record:

DOI: 10.7910/DVN/IUMEDG

Please consult the dataset record for its original data terms and attribution requirements.

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Citation

If you use C-LNO or the associated methodology in research, please cite the project repository and the underlying dataset.

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License

This model repository is released under the MIT License.

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Status

C-LNO is a research model and experimental scientific-machine-learning artifact.

The reported results are intended to document the validated behavior of the model under the evaluation protocols described above.