12 problems

The discussion concerns physics-informed neural networks minimizing a penalty involving the strong PDE residual, while ReLU-based Barron approximations generally provide…

The energy adaptive method is developed for the Ginzburg–Landau energy. Gradient-flow conjecture. Its use on other gradient flow problems could alleviate issues for other complex c…

PINN alternative conjecture. For higher-dimensional problems where finite-difference schemes are no longer applicable, the PINN approach will be a good alternative for solving JCtr…

PINNs are being used to solve the bounded-domain Poisson problem on $\Omega=[-1,1]$, with the manufactured solution $u(x)=\sin(3\pi x)$ and boundary penalty terms. Frequency-bias c…

Optimization-challenge conjecture. When $beta$ is relatively small, so that the solution is smoother, a small $N$ can achieve good accuracy, whereas a larger $N$ may lead to greate…

Consider a family of parameterized partial differential equations and their solutions, where different members are specified by different PDE parameter settings. Problems with simi…

PINN posterior rate conjecture. We conjecture that the PINN posterior distribution actually converges faster than the nonparametric minimax rate.

Let $\Omega$ be the domain, let $\gamma$ specify the $L_\gamma(\Omega)$ norm, let $\mathcal U$ be a model class of functions, and let $\Sigma_n(M)$ be the class of neural-network a…

Let $\mathbf{W}_1$ denote the first-layer weight matrix of the neural network, and let the differential operators be those defining the physics-informed problem. The conjecture. Th…

Pre-training and transfer-learning conjecture. Pre-training and transfer learning are effective because the model is initialized from a better state that produces less noisy predic…

Let $\mathbf{N}$ be a collocation point set generated by a sampling method with low-discrepancy or local-regularity properties, and let the computational domain be the domain on wh…

Infinite-width deterministic-kernel conjecture. For any linear or nonlinear partial differential equation, the NTK of the PINN converges to a deterministic kernel and remains const…