Neural Chains and Discrete Dynamical Systems
This work provides insights into the trade-offs between traditional numerical methods and machine learning for solving PDEs, though it is incremental and limited to one-dimensional problems.
The paper compares neural chains (transformers without self-attention) to discrete dynamical systems, analyzing numerical solutions of Burgers and Eikonal equations via standard discretization and PINNs, finding both methods yield similar knowledge but PINNs use more parameters and lack transparency.
We inspect the analogy between machine-learning (ML) applications based on the transformer architecture without self-attention, {\it neural chains} hereafter, and discrete dynamical systems associated with discretised versions of neural integral and partial differential equations (NIE, PDE). A comparative analysis of the numerical solution of the (viscid and inviscid) Burgers and Eikonal equations via standard numerical discretization (also cast in terms of neural chains) and via PINN's learning is presented and commented on. It is found that standard numerical discretization and PINN learning provide two different paths to acquire essentially the same knowledge about the dynamics of the system. PINN learning proceeds through random matrices which bear no direct relation to the highly structured matrices associated with finite-difference (FD) procedures. Random matrices leading to acceptable solutions are far more numerous than the unique tridiagonal form in matrix space, which explains why the PINN search typically lands on the random ensemble. The price is a much larger number of parameters, causing lack of physical transparency (explainability) as well as large training costs with no counterpart in the FD procedure. However, our results refer to one-dimensional dynamic problems, hence they don't rule out the possibility that PINNs and ML in general, may offer better strategies for high-dimensional problems.