Physics-informed networks get a coordinate change: CTPINN flattens steep gradients and recovers accuracy on boundary layers

Solving PDEs with neural networks gains a route to lowering problem difficulty: instead of only changing the loss, the architecture or the sampling points, the solution itself is made simpler under a coordinate transformation before the network learns it. Prior PINN work concentrated on loss design, network architecture, training strategies and adaptive sampling, all aimed at improving the network's ability to represent or optimize the solution; reducing the complexity of the solution itself had received little attention in the PINN literature. On a one-dimensional boundary-layer problem, a plain PINN fails to produce accurate predictions when the small parameter is very small, while CTPINN solves the problem with both fixed and adaptive transformation parameters; on a two-dimensional boundary-layer problem, every CTPINN variant outperforms the PINN.

The coordinate transformation has two implementations: an explicit formula written from known solution behavior, and a representation by an invertible neural network, both trained jointly with the solution network. The explicit construction follows the stretching approach from grid generation, building the transformation from majorant functions (an integrable upper bound on the solution gradient) so that the transformed derivative is bounded; the implicit construction targets equations where no explicit transformation is available. The implicit construction is a two-layer MLP made invertible by spectral normalization of its Lipschitz constant, and as a plug-in it improves the corresponding baselines on Allen-Cahn, Korteweg-De Vries, lid-driven cavity flow, Helmholtz, Klein-Gordon, Navier-Stokes and high-dimensional Sine-Gordon benchmarks.

The transformation module costs little: the explicit construction adds only one forward pass, and the implicit construction adds moderate training time in exchange for clear accuracy gains. An ablation shows that simply enlarging the network to a comparable parameter count does not deliver the same accuracy improvement as adding the transformation module. All experiments ran on a single NVIDIA A40 GPU with JAX, reporting trainable parameter counts and total wall-clock training time, with the extra time staying moderate relative to the baseline from low- to high-dimensional problems.

The demonstrated scope is one- and two-dimensional boundary-layer problems, a two-dimensional Poisson equation, the Allen-Cahn, Korteweg-De Vries and lid-driven cavity flow equations from the JAX-PI benchmark, the Helmholtz, Klein-Gordon and Navier-Stokes three-dimensional equations from the SPINN benchmark, and high-dimensional Sine-Gordon equations at varying dimension; the explicit construction relies on prior knowledge of solution behavior, and the implicit construction is a two-layer MLP integrated into existing PINN models. The authors propose as next steps designing more suitable transformation functions and mapping irregular domains to regular ones while making solution gradients more uniform.

The authors note neither construction is perfect: explicit transformations rely on prior knowledge of the problem, while implicit constructions lack proper normalization and theoretical guarantees; an adaptive coordinate transformation may also conflict with an adaptive sampling strategy, since both change the solution distribution during training.