Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to use a convergent method in the limit that the grid spacing $\Delta x$ and timestep $\Delta t$ approach zero. Machine learned solvers, which learn to update the solution at large $\Delta x$ and/or $\Delta t$, can never guarantee perfect accuracy. Some amount of error is inevitable, so the question becomes: how do we constrain machine learned solvers to give us the sorts of errors that we are willing to tolerate? In this paper, we design more reliable machine learned PDE solvers by preserving discrete analogues of the continuous invariants of the underlying PDE. Examples of such invariants include conservation of mass, conservation of energy, the second law of thermodynamics, and/or non-negative density. Our key insight is simple: to preserve invariants, at each timestep apply an error-correcting algorithm to the update rule. Though this strategy is different from how standard solvers preserve invariants, it is necessary to retain the flexibility that allows machine learned solvers to be accurate at large $\Delta x$ and/or $\Delta t$. This strategy can be applied to any autoregressive solver for any time-dependent PDE in arbitrary geometries with arbitrary boundary conditions. Although this strategy is very general, the specific error-correcting algorithms need to be tailored to the invariants of the underlying equations as well as to the solution representation and time-stepping scheme of the solver. The error-correcting algorithms we introduce have two key properties. First, by preserving the right invariants they guarantee numerical stability. Second, in closed or periodic systems they do so without degrading the accuracy of an already-accurate solver.

翻译：机器学习偏微分方程求解器以牺牲标准数值方法的可靠性为代价，换取潜在的计算精度和/或速度提升。求解器保证输出精确解的唯一途径，是在网格间距$\Delta x$和时间步长$\Delta t$趋近于零的极限条件下使用收敛方法。而学习在大$\Delta x$和/或$\Delta t$下更新解的机器学习求解器，永远无法保证完美精度。误差不可避免，因此核心问题变为：如何约束机器学习求解器，使其产生的误差符合我们的容忍范围？本文通过保持底层偏微分方程连续不变量的离散对应形式，设计出更可靠的机器学习求解器。这类不变量的典型例子包括质量守恒、能量守恒、热力学第二定律和/或非负密度。我们的核心见解简单明了：为保持不变量，在每一步时间推进中向更新规则施加误差校正算法。虽然此策略不同于标准求解器保持不变量的方式，但它能保留机器学习求解器在大$\Delta x$和/或$\Delta t$下保持精度的灵活性。该策略适用于任意几何构型、任意边界条件下任意时间相关偏微分方程的自回归求解器。尽管此策略极具普适性，但具体误差校正算法需要根据底层方程的不变量特征、解的表达形式以及求解器的时间推进方案进行定制设计。我们引入的误差校正算法具有两个关键特性：第一，通过保持正确的不变量保障数值稳定性；第二，在封闭或周期性系统中，不会降低原本精确求解器的精度。