An existence result is presented for the dynamical low rank (DLR) approximation for random semi-linear evolutionary equations. The DLR solution approximates the true solution at each time instant by a linear combination of products of deterministic and stochastic basis functions, both of which evolve over time. A key to our proof is to find a suitable equivalent formulation of the original problem. The so-called Dual Dynamically Orthogonal formulation turns out to be convenient. Based on this formulation, the DLR approximation is recast to an abstract Cauchy problem in a suitable linear space, for which existence and uniqueness of the solution in the maximal interval are established.

给出了随机半线性演化方程的动态低秩(DLR) 近似的存在性结果。DLR 解决方案通过确定性基函数和随机基函数的乘积的线性组合来逼近每个时刻的真实解，这两者都随时间演变。我们证明的一个关键是找到原始问题的合适等价公式。事实证明，所谓的双动态正交公式很方便。基于这个公式，DLR 近似被重铸为一个合适的线性空间中的抽象柯西问题，为此建立了最大区间内解的存在性和唯一性。