Enforcing governing equation constraints in neural PDE solvers via training-free projections

TL;DR

This work addresses the problem that neural PDE solvers often violate governing equation constraints. It introduces two training-free post hoc projections to enforce the discretized constraint $h(u)=c$: a nonlinear optimization-based projection solved by LBFGS on \min_u \|u-\hat{u}\|+\lambda\|h(u)-c\|, and a linearization-based projection using $h(u)\approx h(\hat{u})+J_h(u-\hat{u})$ to form a linear system $\mathcal{C}u=b$ with exact or relaxed solutions. Evaluation on Lorenz 63, Kuramoto–Shivashinsky, and 2D Navier–Stokes shows substantial reductions in constraint residuals and improved MSE compared with physics-informed baselines, with LBFGS providing the most robust constraint satisfaction and, notably for NS, recovering fine-scale structure. These results highlight the utility of training-free constraint enforcement for temporal-coherent PDE solutions, while also pointing to limitations around differentiability for end-to-end training and the potential for integrating differentiable projections with generative models and SQP-like relinearization strategies.

Abstract

Neural PDE solvers used for scientific simulation often violate governing equation constraints. While linear constraints can be projected cheaply, many constraints are nonlinear, complicating projection onto the feasible set. Dynamical PDEs are especially difficult because constraints induce long-range dependencies in time. In this work, we evaluate two training-free, post hoc projections of approximate solutions: a nonlinear optimization-based projection, and a local linearization-based projection using Jacobian-vector and vector-Jacobian products. We analyze constraints across representative PDEs and find that both projections substantially reduce violations and improve accuracy over physics-informed baselines.

Figures (2)

• Figure 1: NS predictions comparing a PINO baseline with projection methods. Column 1 shows the initial condition $x_0$, Column 2 shows the ground truth $x_T$, and Columns 3-6 show predictions from PINO and projection methods. Only LBFGS successfully recovers the fine-scale structures; physics-informed models fail to capture details they never encountered during training.
• Figure 2: Constraint violation along LBFGS optimization path $u_{0:200}$ from initial guess $u_0 = \hat{u}$ to final solution $u^* = u_{200}$ (black, log scale), where the $x$-axis measures the size $d(k) = \sum_{i=0}^k \|u_{i+1} - u_i \|$ of the path $u_{0:k}$, for $k=0$ to $200$. Blue and red curves show 1st and 2nd order Taylor approximations of the violation. The curves reveal two regimes: linear approximations work well initially but become inaccurate near the solution, while quadratic approximations remain reliable throughout, explaining why LBFGS outperforms linear projection methods.