Low-rank approximation using time-dependent bases (TDBs) has proven effective for reduced-order modeling of stochastic partial differential equations (SPDEs). In these techniques, the random field is decomposed to a set of deterministic TDBs and time-dependent stochastic coefficients. When applied to SPDEs with non-homogeneous stochastic boundary conditions (BCs), appropriate BC must be specified for each of the TDBs. However, determining BCs for TDB is not trivial because: (i) the dimension of the random BCs is different than the rank of the TDB subspace; (ii) TDB in most formulations must preserve orthonormality or orthogonality constraints and specifying BCs for TDB should not violate these constraints in the space-discretized form. In this work, we present a methodology for determining the boundary conditions for TDBs at no additional computational cost beyond that of solving the same SPDE with homogeneous BCs. Our methodology is informed by the fact the TDB evolution equations are the optimality conditions of a variational principle. We leverage the same variational principle to derive an evolution equation for the value of TDB at the boundaries. The presented methodology preserves the orthonormality or orthogonality constraints of TDBs. We present the formulation for both the dynamically bi-orthonormal (DBO) decomposition as well as the dynamically orthogonal (DO) decomposition. We show that the presented methodology can be applied to stochastic Dirichlet, Neumann, and Robin boundary conditions. We assess the performance of the presented method for linear advection-diffusion equation, Burgers' equation, and two-dimensional advection-diffusion equation with constant and temperature-dependent conduction coefficient.

翻译：使用基于时间的基数( TDBs) 的低端近似近距离使用基于时间的基数( TDBs) 已证明对于对随机的硬化部分偏差方程( SPDEs) 进行降序建模是有效的。 在这些技术中, 随机的字段分解成一组确定性的理事会和根据时间的随机的随机的随机的随机的偏差方差方程( SPDEs) 。 当以非均匀的随机的随机的离差基( TDBs) 来应用时, 必须为每届理事会指定适当的 BCs 。 然而, 确定理事会的分数并非微不足道, 因为 (一) 随机的分数分数分数的分数的分数比值值不同; (二) 大多数配数的制式字段的转盘必须保存正正态性或正统的分数限制。 (我们利用的是当前SDRBS- 的分数法 ) 的进化方法, 以当前正数法 或正态的变式的分数法 来评估当前 。