Studying first passage problems using neural networks: A case study in the slit-well microfluidic device

This study presents deep neural network solutions to a time-integrated Smoluchowski equation modeling the mean first passage time of nanoparticles traversing the slit-well microfluidic device. This physical scenario is representative of a broader class of parametrized first passage problems in which key output metrics are dictated by a complicated interplay of problem parameters and system geometry. Specifically, whereas these types of problems are commonly studied using particle simulations of stochastic differential equation models, here the corresponding partial differential equation model is solved using a method based on deep neural networks. The results illustrate that the neural network method is synergistic with the time-integrated Smoluchowski model: together, these are used to construct continuous mappings from key physical inputs (applied voltage and particle diameter) to key output metrics (mean first passage time and effective mobility). In particular, this capability is a unique advantage of the time-integrated Smoluchowski model over the corresponding stochastic differential equation models. Furthermore, the neural network method is demonstrated to easily and reliably handle geometry-modifying parameters, which is generally difficult to accomplish using other methods.

Figure 1

A schematic of electrophoretic sorting of particles by size in the slit-well device. (a) A weak field causes small particles (red) to traverse the device more quickly on average. (b) A strong field causes large particles (blue) to traverse the device more quickly on average.

Figure 2

Schematic of a single periodic subunit of the slit-well device to illustrate passage time models used in this study. (a) Particles are initialized in the left slit (blue particles), confined in the device via a WCA potential on gray walls, undergo Brownian dynamics in the yellow interior (trajectories denoted by black lines) until escaping device at the purple boundary. Dotted line denotes the region of the interior that cannot be occupied by the center of mass of particles. (b) A PDE model of the escape process where the solution $\rho$ satisfies the Smoluchowski equation in the yellow interior. The initial Gaussian band source is located in the left slit, an absorbing boundary in the right slit (purple), and no flux conditions applied on gray walls that move inward to the dotted line to model particle size. (c) A PDE model of the escape process where the solution $g_0$ satisfies the $g_0$ equation (first moment) in the yellow interior region. An absorbing boundary is applied at the right slit wall (purple), and no flux conditions are applied on gray boundaries which move inward to the dotted line to model particle size..

Figure 3

Fully connected feedforward neural network of width $w$ and depth $d$ mapping coordinates $(x,y)$ and problem parameters $(\lambda ,\sigma )$ to an output ${\tilde{g}}_0(x,y;\lambda ,\sigma )$. At each node, a weighted sum of the incoming arrows and a bias is computed and passed through an activation function. The network's parameters are optimized such that ${\tilde{g}}_0(x,y;\lambda ,\sigma )$ approximately satisfies the target PDE and BCs.

Figure 4

NNM solutions to the $g_0$ equation subject to (a) a small particle size and a weak electric field, (b) a small particle size and a strong electric field, (c) a large particle size and a weak electric field, and (d) a large particle size and a strong electric field.

Figure 5

Analysis of $g_0$ solutions computed using the NNM. Mean first passage times $\langle \tau \rangle$ for the NNM (a) with fixed parameters, and parametrized by (b) $\lambda$, (c) $\sigma$, and (d) both $\lambda$ and $\sigma$. Star markers denote values obtained using the FEM, and insets display behavior at high field strengths. Effective mobilities ${\mu }_{\mathrm{eff}}$ for the NNM (e) with fixed parameters, and parametrized by (f) $\lambda$, (g) $\sigma$, and (h) both $\lambda$ and $\sigma$. Star markers denote values obtained using FEM, and insets zoom in on the minimum of the curves. Relative errors $\varepsilon$ computed against the FEM for the NNM (i) with fixed parameters, and parametrized by (j) $\lambda$, (k) $\sigma$, and (l) both $\lambda$ and $\sigma$. Dotted black line denotes 1% error baseline computed by particle simulations. (m) Testing loss of the NNM with fixed parameters, and marginal loss of the NNM parametrized by (n) $\lambda$, (o) $\sigma$, and (p) both $\lambda$ and $\sigma$.

Figure 6

Analysis of the $g_0(x,y;\lambda ,\sigma )$ solution obtained using the NNM parametrized by both field strength $\lambda$ and particle size $\sigma$. (a), (e) Mean first passage time $\langle \tau \rangle$ with white contours to show nonmonotonic sorting behavior. (b), (f) Effective mobility ${\mu }_{\mathrm{eff}}$ with white contours to show saddle point. (c), (g) Relative error $\varepsilon$ of the MFPT with white contour to denote 1% error threshold. (d), (h) Marginal loss $\mathcal{L}\left(g_0\right|\lambda ,\sigma )$ with white contour to denote testing loss $\mathcal{L}$.

Figure 7

Contour plots of the baseline electric potential $u_0$ (black) and field ${\vec{E}}_0$ (red) computed by the NNM.

Figure 8

Passage time properties of particles escaping the slit-well model computed using Brownian dynamics simulations. Star markers denote values obtained via FEM. (a) Mean first passage time $\langle \tau \rangle$. (b) Effective mobility ${\mu }_{\mathrm{eff}}$. (c) Relative error of $\langle \tau \rangle$ computed against the ground truth FEM solution.