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，换取精度和/或速度上的潜在提升。求解器保证输出精确解的唯一途径，是采用在网格间距$\Delta x$和时间步长$\Delta t$趋于零时收敛的方法。而机器学习求解器通过学习在较大$\Delta x$和/或$\Delta t$下更新解，永远无法保证完美精度。一定程度的误差在所难免，因此问题变为：如何约束机器学习求解器，使其产生的误差是我们愿意容忍的类型？本文通过保持底层偏微分方程连续不变量的离散模拟，来设计更可靠的机器学习求解器。此类不变量的示例包括质量守恒、能量守恒、热力学第二定律和/或非负密度。我们的关键见解很简单：为保持不变量，在每一时间步对更新规则应用误差修正算法。尽管此策略不同于标准求解器保持不变量的方式，但它对于保留机器学习求解器在较大$\Delta x$和/或$\Delta t$下保持精度的灵活性是必要的。该策略可应用于任意几何形状、任意边界条件下任意时间相关偏微分方程的任何自回归求解器。尽管此策略非常通用，但具体的误差修正算法需要根据底层方程的不变量以及求解器的解表示和时间步进方案进行定制。我们引入的误差修正算法具有两个关键性质：首先，通过保持正确的不变量，它们保证了数值稳定性；其次，在封闭或周期系统中，它们能在不降低已有精确求解器精度的情况下实现这一目标。