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Mathematical Foundations
Mathematical Foundations
The math behind each layer — PINNs, Grad-Shafranov, MPC, GNNs, state estimation — anchored to Kronos's use.
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The math behind each layer — PINNs, Grad-Shafranov, MPC, GNNs, state estimation — anchored to Kronos's use. Every page in this section is part of the same control stack — click any card to go deeper, or return to the Master Blueprint to see how it connects.
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Ambipolar Potential and Pastukhov ConfinementBayesian Inference FoundationsBayesian Optimization for Scenario DesignBootstrap Current and Profile Self-ConsistencyConstrained Optimization, Lagrangians, and DualityEquilibrium Reconstruction as an Inverse ProblemEquivariance and Invariance in Graph NetworksExtended and Unscented Kalman FiltersFlux Coordinates and the Safety FactorFree-Boundary Grad-ShafranovGaussian ProcessesGraph Neural Networks for Coupled SubsystemsGraph Neural Networks: Message-Passing MathematicsGreen's Functions and Poloidal-Field Coil ResponseInformation Theory for Sensor SelectionMCMC and Hamiltonian Monte CarloMPC Constraints and the KKT ConditionsMPC QP Solver MathematicsMathematical Foundations of the S.M.A.R.T. GeneratorMirror Interchange Stability and Minimum-BModel-Predictive Control: The Optimal-Control FormulationMonte Carlo and Variance ReductionNegative-Triangularity Shaping MathematicsNeural Operators: DeepONet and Fourier Neural OperatorsObservability and State EstimationPINN Conditioning and Training DynamicsPINN Formulation for Grad-ShafranovPINN Loss Construction and WeightingPINN for the Burner Ambipolar PotentialPINNs for Inverse ProblemsPhysics-Informed Neural Networks: FundamentalsReceding Horizon, Terminal Cost, and StabilityResistive Wall Mode MathematicsSpectral Graph Theory for DiagnosticsStability-Problem Map Across Both MachinesTandem-Mirror Equilibrium MathematicsThe Ballooning Mode Equation and s-alpha SpaceThe Ensemble Kalman FilterThe Grad-Shafranov EquationThe Ideal MHD Energy PrincipleThe Kalman FilterThe Linear MHD Stability EigenproblemThe Linear-Quadratic Regulator BaselineTrajectory Optimization for Scenario DesignVariational Inference for Fast UncertaintyVertical Stability MathematicsWeak Form and Finite-Element Discretization of Grad-Shafranov