The Physics Is the Network: Fast, Training-Free Neural Emulation
Authors: Luka Todorovic, Kaushal Gianchandani, Sai Ravela
Organizations: Department of Mathematics and Department of Statistics, University of Michigan, Ann Arbor, MI · NASA Postdoctoral Program Fellow, NASA Goddard Institute for Space Studies, New York, NY · Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA
Neural emulation need not require learning a substitute for known physics. We investigate PolyNets that couple a continuous-time neural representation of polynomial dynamics with a simple recurrent network for discrete-time integration. When the governing coefficients are known, the resulting tensor graph requires no training and preserves the prescribed numerical scheme while enabling efficient hardware execution and parallel simulation. Using the one-dimensional Korteweg-de Vries equation, we compare PolyNet with CPU and GPU finite-difference Runge-Kutta solvers and three trained neural models: a state-map emulator, a physics-informed predictor, and a learned-tendency integrator. Analytical soliton solutions provide a reference for amplitude and phase errors across grid resolutions, 10^6 dispersively scaled time steps, ensembles of up to 10^4 trajectories, and varying initial amplitudes. PolyNet achieves accuracy comparable to the numerical solvers, including at amplitudes outside the learned models' training conditions. It combines this fidelity with high GPU throughput, approximately constant over the tested resolution range, and favorable ensemble scaling.
Figures & tables
Figure 1: PolyNet graph for the semidiscrete KdV tendency, FN(U)=−6U⊙D1U−D3U . Linear derivative branches and a componentwise product implement the prescribed physics. The four-stage recurrent RK4 time integrator is not shown.
Figure 2: Training time for the three learned models and setup time for CPU finite-difference RK4, GPU finite-difference RK4, and PolyNet versus grid size N . The unsuccessful NPDE configuration at N=100 is omitted. Shading around log–log fits includes an imposed minimum display width and should not be interpreted as a confidence interval.
Figure 3: Time-step throughput versus grid size N . Each timed block advances 1000 steps; throughput in steps/ms is 1000/Tmedian , where Tmedian is the median block duration in milliseconds. Shading around log–log fits includes an imposed minimum display width and is not a confidence interval.
Figure 4: Maximum sampled absolute (a) phase and (b) amplitude errors against the analytical solitary-wave reference versus grid size N , over 106 steps at the prescribed resolution-dependent dispersive time step. Unsuccessful GPU finite-difference RK4 configurations at N=100 and N=200 are omitted.
Figure 5: Median batch-step throughput (steps/ms) versus the number of independent soliton trajectories at N=400 . Each batch step advances the entire ensemble; aggregate trajectory-step throughput is the plotted value multiplied by ensemble size.
Figure 6: Maximum sampled absolute (a) phase and (b) amplitude errors versus initial soliton height h , for N=400 and 106 integration steps. The neural models were trained at h=3 using analytical soliton targets; other heights were held out. Periodic phase separation is bounded by L/2=20 , explaining the saturated phase-error curves.
Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.