Energy dissipation moves from penalty term to network structure: EDP-PI-DOSnet keeps long-time predictions stable without retraining
Synopsis
EDP-PI-DOSnet embeds classical energy-stable schemes into a physics-informed operator-splitting network so that the learned discrete operators satisfy a discrete energy dissipation law. The earlier PI-DOSnet did not guarantee that its learned linear operator was symmetric negative semidefinite, energy could rise between steps, and long-time extrapolation needed two or three rounds of retraining. On the Allen–Cahn equation, both EDP-PI-DOSnet variants keep the modified energy non-increasing at every observed time level and give smaller long-time prediction errors without retraining.
Interpretation
The new network guarantees discrete energy dissipation by construction rather than approximating it through an energy penalty in the loss. The original PI-DOSnet did not guarantee a symmetric negative semidefinite learned linear operator, and its operator splitting did not guarantee energy dissipation, so energy could grow between consecutive steps. On the Allen–Cahn equation, the learned operators of EDP-PI-DOSnet and of the second-order central finite difference scheme have all eigenvalues on the non-positive real axis, whereas those of PI-DOSnet and its penalty variant spread across the left half of the complex plane.
The construction constrains convolution kernels so the learned linear operator is equivalent to a symmetric negative semidefinite matrix, then places semi-implicit time schemes inside each time block. The constraint uses the fact that convolution with circular padding is equivalent to a circulant matrix, and a corollary of the Gershgorin circle theorem pushes the spectrum onto the non-positive real axis. Networks are built on the Crank–Nicolson, SAV/CN and R-DVD-1 schemes and trained and recursively predicted on the Allen–Cahn equation and the mass-conserved Allen–Cahn–Navier–Stokes two-phase flow case.
Long-time prediction no longer requires retraining, a direct consequence of the structural guarantee. The original PI-DOSnet, trained on one time interval and extrapolated to a longer one, typically needed two or three rounds of retraining to reach the desired accuracy. With large-step inference advancing one time unit per step, the R-DVD-1 and SAV/CN variants of EDP-PI-DOSnet have nearly overlapping error curves, reach prediction errors of about the same order at the reported time, and are trained only once.
Perspective
The demonstrated scope is the one-dimensional Allen–Cahn equation and the mass-conserved Allen–Cahn–Navier–Stokes two-phase flow on a periodic domain, with reference solutions from Chebfun and the corresponding reference schemes; networks are trained on a training interval and then recursively extrapolated to examine long-time behavior.
The network guarantees dissipation of a modified energy, which is not necessarily identical to the original energy, so monotonic decrease of the original energy at every time step is not theoretically guaranteed; the learnable extrapolation strategy makes the original energy evolution smoother, but strict monotonicity is likewise not proved.