Tensor-Train Layers for Physics-Informed Neural Networks
Abstract
Abstract Physics-informed neural networks (PINNs) embed partial-differential-equation (PDE) constraints directly into the loss function, enabling mesh-free solution of forward and inverse problems. Their deployment on resource-constrained platforms, however, is hindered by the large parameter counts of wide networks required to resolve sharp solution features. We study tensor-train (TT) layers as a compression mechanism for the dominant hidden-to-hidden maps in PINNs, reducing parameter counts by up to 8.8× while retaining 97% of the solution accuracy of the dense baseline. The method combines TT-SVD initialization from a pre-trained dense network with a per-step weight cache that avoids repeated TT-core contractions during the higher-order automatic-differentiation passes required by the PDE residual. We evaluate the resulting TT-PINN models on two benchmark PDEs of contrasting regularity: the one-dimensional viscous Burgers equation and the Black-Scholes option-pricing equation. Across both problems, the bond dimension χ controls a clear accuracy-compression trade-off, while the smoother Black-Scholes solution remains systematically more compressible than the shock-dominated Burgers solution at fixed χ. These results position TT layers as a practical low-rank parameterization for PINNs and show that solution regularity is a key determinant of compression quality in scientific machine learning surrogates. 2020 AMS Subject Classification: 68T07, 65M99, 15A69, 35Q68, 91G60
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