Optimal Transport Neural Operator

• Optimal Transport Neural Operator (OTNO) integrates optimal transport theory with neural operator learning for enhanced calibration and model fidelity.
• The framework employs structure functionals, using metrics like the Earth Mover's Distance, to diagnose and correct smooth model errors against noise.
• OTNO demonstrates practical benefits in applications such as medical and seismic imaging by improving operator calibration through data-driven learning.

Optimal Transport Neural Operator (OTNO) frameworks integrate the mathematical structure of optimal transport (OT)—the theory of transporting one probability measure onto another with minimal cost—into neural operator learning. The OTNO paradigm leverages both deep learning and the geometric/variational structure of OT to model transport between complex data distributions, with applications ranging from generative modeling and PDE solving to robust domain adaptation and geometric representation. Depending on the implementation, the OT machinery may be used as a post-hoc diagnostic, a regularized loss, a core building block for stochastic map parameterization, or as part of the neural architecture itself. This entry reviews key principles, methodologies, and connections in OTNOs, drawing on recent research including (Puthawala et al., 2018).

1. OT Foundations and Diagnostic Structure Functionals

Optimal transport is grounded in the minimization problem

$$W_p(\rho_1,\rho_2) = \left( \min_{\pi} \int_{\Omega\times\Omega} c(x^1, x^2)^p \pi(x^1, x^2) \,dx^1 dx^2 \right)^{1/p},$$

where $c(\cdot,\cdot)$ is a ground metric (typically Euclidean norm), and $\pi$ is a coupling with prescribed marginals. The $p=1$ case, called the Earth Mover's Distance (EMD), is especially prevalent and can be reformulated in the continuum as

$$EMD(\rho_1, \rho_2) = \min_m \int_\Omega |m(x)|_2 dx \quad \text{subject to}\quad \nabla\cdot m(x) + \rho_2(x) - \rho_1(x) = 0.$$

(Puthawala et al., 2018) introduces the structure functional as a diagnostic based on the EMD. For any $f \in L^1(\Omega)$: $struc(f) = EMD(f^+, f^-)$ where $f^+(x) = \max(f(x)-\mu,0)$, $f^-(x)=\max(\mu-f(x),0)$, and $\mu = \frac{1}{|\Omega|}\int_\Omega f(x)dx$.

The key property is that $struc(f)$ is sensitive to the "smoothness" or spatial correlation of $f$; white noise yields low values whereas structured model errors produce higher values, thus distinguishing miscalibration of forward operators from measurement noise in inverse problems.

2. OTNO Methodologies: Diagnostic, Supervised, and Dynamic Flows

There are several pathways for embedding OT into neural operator learning:

• Diagnostic Post-Processing: As in (Puthawala et al., 2018), the OT machinery is applied after a solution is obtained (for example, after minimizing a regularized inverse problem). By evaluating $struc(r)$, with $r$ the residual, one can calibrate operator parameters and even design corrective procedures to improve the forward operator.
• Loss/Training Objective: In other OTNO variants, the Wasserstein distance (or its derivatives, such as Sinkhorn divergences) is used as a training loss to align outputs of neural operators with target measures. This is akin to optimal transport being a "shape-aware" distance between predicted and ground-truth fields.
• Supervised Learning with OT Oracle: As in (Schioppa, 2019), fast discrete OT solvers (e.g., Sinkhorn) can construct "ground-truth" or proxy transport plans or potentials on minibatches. Neural networks are then trained to regress these OT plans, learning fast parametric approximations.
• Dynamic Flow Alignment: Another method (Schioppa, 2019), representing the transport map as a neural network $T_w$ and evolving $w$ via gradient descent on a Lagrangian which combines transport cost and a discrepancy term enforcing pushforward constraints. This synthetic flow can be monitored to inspect the convergence and geographic faithfulness of the learned operator.

3. Structure Functionals for Operator Error Diagnosis and Correction

The structure functional $struc(f)$ has both theoretical and applied significance:

Functional	Definition	Sensitivity
$struc(f)$	$EMD(\max(f-\mu,0), \max(\mu-f,0))$	High for smooth/model errors, low for noise

• Diagnostic Role: Low values of $struc(r)$ indicate noise-dominated residuals; high values signal misspecified system operators.
• Corrective Role: In some inverse problems, adjusting the operator parameters to minimize $struc(r)$ can guide calibration toward the true operator, outperforming standard $L^1$/$L^2$-norm–based approaches for residual analysis.
• Robustness: The semi-norm is shift-invariant and homogeneous, making it well-suited for large classes of inverse problems under regularization.

In applications such as seismic or medical imaging, where the forward model is only partially known, embedding $struc(r)$ as a layer or wrapper in a neural operator could facilitate continuous model correction.

4. OTNO Versus Standard Neural Operators

The principal distinction between the use of optimal transport in (Puthawala et al., 2018) and in generic OTNO frameworks is methodological:

• OT as Standalone Diagnostic (as in (Puthawala et al., 2018)):
  • OT-based diagnostics are applied externally, after solving the forward/inverse problem, to measure and correct structured error in the residual.
  • The approach is agnostic to the architecture—any neural or classical solver could be wrapped with this structure-based diagnostic.
  • Correction via "structure plans" is feasible in low-parameter problems.
• OT as Integrated Learning Principle (OTNOs):
  • The operator itself is parameterized as a neural network and trained via losses based on Wasserstein metrics, or is designed to represent the transport map (static or dynamic).
  • In this setting, OT is not just diagnostic but generative (maps between functional spaces are learned end-to-end).
  • For large-scale or high-dimensional function spaces, the dynamic flow or supervised OT reduction pathways enable scalable, data-driven learning.

The structure diagnostic could, in principle, be incorporated as a calibration tool within the training loop of a neural operator, particularly to enforce physical consistency or model fidelity.

5. Applications and Empirical Results

Numerical experiments in (Puthawala et al., 2018) indicate:

• Inverse Problems: For scalar or vector-parametrized forward operators (e.g., $L_\theta$), calibration based on minimizing $struc(r_{\theta,\eta})$ achieves sharper minima and greater sensitivity than classical residual norms.
• Sensitivity to Operator Parameters: The structure functional is highly responsive to model miscalibration, thus enabling operator identification amidst measurement noise.
• Generalizability: The structure-based approach is highlighted for medical imaging, seismic imaging, and plasma diagnostics, where operator calibration is essential.

This use case is complementary to the broader OTNO framework, which aims at end-to-end operator learning but can profit from such diagnostic principles for robust deployment.

6. Theoretical and Practical Implications

The deployment of structure-based EMD functionals provides several notable outcomes:

• Distinct Attribution of Error: Unlike classical norms, the structure semi-norm distinguishes smooth/model error from rough/noise error at the residual level, enabling targeted correction.
• Potential for Integration: While originally post-hoc, structure diagnostics could be integrated within learning-based operator schemes, particularly those leveraging OT losses or mappings in high dimensions.
• Computational Feasibility: Fast algorithms for the EMD make such diagnostic tools practical for real-world, high-dimensional applications.

A plausible implication is that the systematic use of structure metrics could enhance model transparency and allow for both automatic operator tuning and the quantification of uncertainty due to modeling error in neural operator frameworks.

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In summary, the diagnostic and corrective use of optimal transport theory as exemplified in (Puthawala et al., 2018) provides a rigorous, sensitive, and computationally viable pathway to measure and improve the fidelity of forward operators, especially within regularized inverse problems. While OTNO architectures typically integrate transport theory into the learning pipeline, the structure approach offered by (Puthawala et al., 2018) serves as a robust, operator-agnostic wrapper that can complement and enhance the utility of OTNOs in both classic and data-driven inverse modeling.