Finite-Width NTK Digital Twins: Closed-Form Uncertainty Quantification for Sustainable Circular Economy Control
Keywords:
Digital Twin, Circular Economy, Physics-Informed Neural Operator, Neural Tangent Kernel (NTK), Uncertainty QuantificationAbstract
We introduce a digital twin framework for circular economy systems that replaces conventional black-box surrogates and Monte-Carlo sampling with a closed-form Gaussian process derived from the finite-width Neural Tangent Kernel (NTK) of a physics-informed neural operator. The core methodology involves training a Fourier neural operator under physics-informed residual losses that enforce mass balance, thermodynamic entropy constraints, and recycling efficiency targets. After training, we extract the empirical NTK and augment it with Jacobian blocks from the PDE residuals, yielding an analytically tractable predictive distribution over twin states. This approach provides exact epistemic uncertainty maps for key circularity metrics, such as the circularity index and embodied carbon, without any sampling. The predictive mean and covariance are then fed into a model-predictive control module that optimizes actions—for example, rerouting waste streams or adjusting recycling temperatures—by analytically computing the expected cost under the Gaussian posterior. A hyperparameter optimization module further adapts the physics-informed weights and kernel combination weights via stochastic marginal-likelihood gradients, enabling rapid adaptation to new lifecycle data without retraining the neural operator from scratch. The system integrates with conventional IoT sensor feeds and ERP systems through a learned encoder that maps sensor readings to initial conditions and boundary conditions for the operator. Our main contribution is a unified, sampling-free uncertainty quantification mechanism that operates in real time and respects the underlying physical laws of material loops and reverse logistics. This work demonstrates that finite-width NTK theory can be practically deployed for risk-aware control of closed-loop value chains, offering a principled alternative to heuristic uncertainty estimation in sustainable engineering.
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