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Challenges in Training PINNs: A Loss Landscape Perspective
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Theoretical analysis of why PINN training is hard: differential operators make the Gauss-Newton matrix ill-conditioned (with spectral decay tied to the operator), and gradient descent plus damped Newton converge locally under a PL-type condition.
Problem
Understanding why training Physics-Informed Neural Networks is difficult, focusing on loss-landscape ill-conditioning caused by differential operators in the PDE residual term.
Approach
Relates the conditioning of the population and empirical Gauss-Newton matrices of the PINN loss to the spectrum of the operator composition A∘K (A = D*D), showing that fast eigenvalue decay implies a large condition number (κ_L ≥ n^β or exp(β n^{1/2})) and hence hard optimization; also gives local linear/damped-Newton convergence guarantees under a PL* condition.
Key results
- Scenario 1 (polynomial eigenvalue decay with beta>=1) gives condition number kappa_L(S) >= C n^beta for any set S containing a minimizer, with high probability.
- Scenario 2 (exponential eigenvalue decay) gives kappa_L(S) >= C exp(beta n^{1/2}).
- Gradient descent with step 1/beta_L converges linearly to a neighborhood of a minimizer under a mu-PL* condition, followed by damped Newton with local convergence rate (2/3)^k.
Limitations
- Results are theoretical; the excerpt contains no experiments validating the bounds.
- Relies on ridge incoherence and PL* assumptions whose verification for PINNs is not shown in the excerpt.
Experimental setup
Builds on: neural tangent kernel PL* condition (Liu, Zhu, Belkin) De Ryck and Mishra eigenvalue result
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