A Low Rank Neural Representation of Entropy Solutions

TL;DR

The paper tackles real-time simulation of entropy solutions to nonlinear scalar conservation laws in one spatial dimension by introducing a Low-Rank Neural Representation (LRNR). It develops a compositional representation of entropy solutions through rarefied and relief characteristics and encodes this structure into a low-rank implicit neural network with time-evolving, but linearly dependent, coefficients. The authors prove that LRNR can uniformly approximate both classical and entropy solutions with error decaying like $O(1/K)$ while using fixed, small ranks (e.g., $r=2$, depth $5$ for entropy), regardless of shock topology. This yields near-constant online evaluation costs per space-time point, enabling fast reduced-order modeling of hyperbolic PDEs and scalable handling of complex shock interactions.

Abstract

We construct a new representation of entropy solutions to nonlinear scalar conservation laws with a smooth convex flux function in a single spatial dimension. The representation is a generalization of the method of characteristics and posseses a compositional form. While it is a nonlinear representation, the embedded dynamics of the solution in the time variable is linear. This representation is then discretized as a manifold of implicit neural representations where the feedforward neural network architecture has a low rank structure. Finally, we show that the low rank neural representation with a fixed number of layers and a small number of coefficients can approximate any entropy solution regardless of the complexity of the shock topology, while retaining the linearity of the embedded dynamics.

Figures (10)

• Figure 1: An example of solution to the Burgers' equation. Time variable has been rescaled to better show details.
• Figure 1: An example of an extended initial condition $\hat{u}_0$ of $u_0$ in the case of Burgers' flux. The newly assigned values of $\hat{u}_0$ in the interval $\Gamma_1$ are given by \ref{['eq:uhat']}. The values are linear due to the choice of the Burgers' flux.
• Figure 1: The time evolution of the rarefied characteristics $\widehat{X}(\cdot, t)$ for the stationary shock example in Fig. \ref{['fig:stationary']}, zoomed into the region relevant to shock propagation. The constant interval (marked by $\Upsilon_t$) expands over time $0 < t_1 < t_2 < t_3$. The two kinks at the endpoints (marked by two arrows) of this interval travel in opposite directions over time.
• Figure 2: An example of the extension $u_0$ to $\hat{u}_0$ as in \ref{['eq:uhat']} for the Burgers flux $F(u) = u^2 / 2$. The function $\hat{I}^+$ has a jump discontinuity of magnitude $|\Gamma_1|$ as depicted in the upper right plot. Due to this jump, the values of $\hat{u}_0$ inside interval $\Gamma_1$ highlighted in the upper left plot are effectively omitted in the composed function $\hat{u}_0 \circ \hat{I}^+$. This effect is shown in the lower right plot.
• Figure 2: An illustration of time-layered approximation $\Xi_k$\ref{['eq:Xik']} immediately after shock formation, compared with the rarefied characteristics with multivalued inverse $\widehat{X}_0$ and the rarefied characteristics $\widehat{X}$. At uniform grid times $t_k$\ref{['eq:tunif']} the $\Xi_k(\cdot, t_k) = \widehat{X}(\cdot, t_k)$, however, as the time is evolved forward $\Xi_k(\cdot, t)$ is different from $\widehat{X}(\cdot, t_k)$ in general, and the difference is within the $\epsilon$ error bound.

Theorems & Definitions (22)

• Lemma 2.1
• Proof 1
• Definition 3.1: Feedforward neural network architecture
• Definition 3.2: Feedforward Neural Networks
• Definition 3.3: Low Rank Neural Representation (LRNR)
• Definition 4.1: Admissible coefficients
• Definition 4.2: Transported subspace
• Lemma 4.3
• Proof 2
• Example 4.4: A 2-layer version of LRNR