Solving BDNK diffusion using physics-informed neural networks

TL;DR

This work reformulates relativistic BDNK diffusion in flux-conservative form and solves it in (1+1)D using both a KT finite-volume scheme and a SA‑PINN‑ACTO framework that algebraically enforces initial and periodic boundary conditions. The SA‑PINN‑ACTO approach combines self-adaptive collocation weights with exact IC/BC enforcement via output transforms, enabling the network to focus on the PDE residual. Across smooth cases, SA‑PINN‑ACTO matches KT accuracy to ~10^-3 in relative $L^2$, while for discontinuous data the PINN exhibits expected smoothing and larger errors, with KT remaining faster and more precise for shock features. The results demonstrate that physics-informed neural networks offer a flexible, differentiable alternative with potential for inverse problems and higher-dimensional extensions in relativistic hydrodynamics.

Abstract

In this work, we reformulate the relativistic BDNK (Bemfica-Disconzi-Noronha-Kovtun) diffusion equation in flux-conservative form, and solve the resulting equations in $(1+1)$D using both a second-order Kurganov-Tadmor finite volume scheme and physics-informed neural networks (PINNs). In particular, we introduce the SA-PINN-ACTO framework, which combines the self-adaptive PINN technique with an exact enforcement of initial and periodic boundary conditions through an algebraic transform of the network's raw output, allowing the network to focus solely on minimizing the PDE residual. We test both approaches on smooth and discontinuous initial data, for both trivial and dynamically evolving velocity and temperature BDNK backgrounds, and for two characteristic speeds. The SA-PINN-ACTO method matches the converged Kurganov-Tadmor solutions for smooth profiles, while for discontinuous profiles the errors increase, reflecting an expected limitation of PINNs near sharp gradients.

Figures (11)

• Figure 1: Vanilla PINN architecture and loss construction for the $(1+1)$D BDNK diffusion problem. The network takes spacetime inputs $(t,x)$ and predicts $(J^0,\alpha)$. It is fundamental that the quantities $J^x$ and $N_0$ are obtained from the constitutive relations from Sec. \ref{['ssec:flux_conservative_formulation']} and not from the identities $N_0=-\partial_t\alpha$ and $\partial_x J^x = - \partial_t J^0$, since that would decouple the output variables $\alpha$ and $J^0$ and incorrectly put them on independent grounds. Afterwards, automatic differentiation (AD) yields $\partial_t J^0$, $\partial_t\alpha$, and $\partial_x J^x$. The residuals $\mathcal{L}_{\rm IC}$, $\mathcal{L}_{\rm BC}$ and $\mathcal{L}_{\rm PDE}$ are calculated as shown in Eqs. \ref{['eq:loss_ic']}-\ref{['eq:loss_pde']}, respectively, and the total loss $\mathcal{L}\equiv\mathcal{L}_{\rm PDE}+\lambda_{\rm IC}\mathcal{L}_{\rm IC}+\lambda_{\rm BC}\mathcal{L}_{\rm BC}$ is minimized by computing $\nabla_\theta \mathcal{L}$ and updating $\theta$ using gradient descent (Adam), followed by quasi-Newton refinement (L-BFGS).
• Figure 2: SA-PINN-ACTO architecture and loss construction for the $(1+1)$D BDNK diffusion problem. As in the vanilla PINN (see Fig. \ref{['fig:bdnk-vanilla-pinn']}), the network takes spacetime inputs $(t,x)$, but unlike it, the SA-PINN-ACTO predicts normalized raw outputs $\hat{J}^0/s_{J^0}$ and $\hat{\alpha}/s_{\alpha}$. These are first denormalized, then passed through the IC-enforcing transform in Eq. \ref{['eq:algebraic_enforcement_of_ICs']}, and then through the BC-enforcing transform in Eq. \ref{['eq:algebraic_enforcement_of_BCs']} (together, the ACTO transform), yielding physical outputs $(J^0,\alpha)$ that exactly satisfy the prescribed initial data and periodic boundary conditions. Automatic differentiation (AD) is then used to compute the normalized PDE residuals $R_{{\rm PDE},i}$ defined in Eq. \ref{['eq:sa_pinn_residuals']}; these are weighted pointwise by the self-adaptive factors $\lambda_i^2$ to build the PDE loss $\mathcal{L}_{\rm PDE}\equiv\mathcal{L}$ in Eq. \ref{['eq:loss_pde_sa']}. The logits underlying $\lambda_i=\mathrm{softplus}(z_i)$ are optimized jointly with the network parameters $\theta$: AD provides $\nabla_{\theta}\mathcal{L}$ to update $\theta$ so as to minimize $\mathcal{L}$ and $\nabla_{\lambda_i}\mathcal{L}$ to adapt the weights, focusing learning on collocation points with larger residuals.
• Figure 3: Results from KT for the first setup. (a) Evolution of $n$ for $c_{\rm ch}=0.5$. (b) Evolution of $n$ for $c_{\rm ch}=0.9$. (c) Evolution of $J^0$ for $c_{\rm ch}=0.5$. Total charge $\int_{-L}^{L}{J^0\,dx}$ conserved at all times up to a fraction of $4.4\times 10^{-15}$ of the initial charge. (d) Evolution of $J^0$ for $c_{\rm ch}=0.9$. Total charge $\int_{-L}^{L}{J^0\,dx}$ conserved at all times up to a fraction of $4.9\times 10^{-15}$ of the initial charge.
• Figure 4: Results from the SA-PINN-ACTO for the first setup. For these simulations, we use $|N_{\rm PDE}|=20,000+500$, and $25,000$ Adam epochs, followed by L-BFGS fine-tuning. (a) Evolution of $n$ for $c_{\rm ch}=0.5$. (b) Evolution of $n$ for $c_{\rm ch}=0.9$. (c) Evolution of $J^0$ for $c_{\rm ch}=0.5$. Total charge $\int_{-L}^{L}{J^0\,dx}$ conserved at all times up to a fraction of $1.1\times 10^{-4}$ of the initial charge. (d) Evolution of $J^0$ for $c_{\rm ch}=0.9$. Total charge $\int_{-L}^{L}{J^0\,dx}$ conserved at all times up to a fraction of $8.6\times 10^{-5}$ of the initial charge. (e) Total loss history for $c_{\rm ch}=0.5$. Best loss was $7.752\times 10^{-10}$. Total training time: $434.64$ seconds (Adam: $411.78$ seconds; L-BFGS: $22.86$ seconds). (f) Total loss history for $c_{\rm ch}=0.9$. Best loss was $1.862\times 10^{-10}$. Total training time: $429.12$ seconds (Adam: $406.95$ seconds; L-BFGS: $22.17$ seconds).
• Figure 5: Results from KT for the second setup. (a) Evolution of $n$ for $c_{\rm ch}=0.5$. (b) Evolution of $n$ for $c_{\rm ch}=0.9$. (c) Evolution of $J^0$ for $c_{\rm ch}=0.5$. Total charge $\int_{-L}^{L}{J^0\,dx}$ conserved at all times up to a fraction of $4.1\times 10^{-16}$ of the initial charge. (d) Evolution of $J^0$ for $c_{\rm ch}=0.9$. Total charge $\int_{-L}^{L}{J^0\,dx}$ conserved at all times up to a fraction of $4.1\times 10^{-16}$ of the initial charge.