Unified generalization bounds for PINNs and VPINNs are derived by representing nonlinear differential operators via Taylor expansion and Koopman theory, showing high-rank networks generalize well while nonlinearity exponentially enlarges the bounds.
Bartlett.Neural network learning: Theoretical foundations
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Derives algorithm-dependent generalization bounds for neural nets using multilevel entropic regularization and proposes a Metropolis-simulated multi-scale Gibbs training procedure tested on a two-layer net for MNIST.
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Unified generalization analysis for physics informed neural networks
Unified generalization bounds for PINNs and VPINNs are derived by representing nonlinear differential operators via Taylor expansion and Koopman theory, showing high-rank networks generalize well while nonlinearity exponentially enlarges the bounds.
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Chaining Meets Chain Rule: Multilevel Entropic Regularization and Training of Neural Nets
Derives algorithm-dependent generalization bounds for neural nets using multilevel entropic regularization and proposes a Metropolis-simulated multi-scale Gibbs training procedure tested on a two-layer net for MNIST.