Learning mean-field equations from particle data using WSINDy

Learning anisotropic interaction rules from individual trajectories in a heterogeneous cellular population

“…At its core, our method involves learning ordinary differential equations for cells using available trajectory data. For this we employ the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy), which has been shown to successfully identify governing equations from data at the levels of ordinary different equations [62], partial differential equations [63], first-order interacting particle systems [12] and even works in a small-memory online streaming scenario [64]. A significant advantage of the WSINDy method is that it identifies a single governing equation which can be interpreted, analysed and simulated using conventional techniques of applied mathematics.…”

“…The linear inequality constraint Cw ≤ d encodes the constraints listed in (2.2), (2.3) and (2.4) on the forces on f a−r , f align and f drag , and λ is the sparsity threshold. We employ the modified sequential thresholding algorithm from [12,63], with least-squares iterations replaced by solving the associated linearly constrained quadratic program. 5 Since the coefficients b w ðiÞ have no a priori absolute magnitude, we threshold only on the magnitudes of the given term relative to the response vector b (i) , namely, we define the thresholding operator H l ðwÞ by A sweep over 40 equally log-spaced λ values l ¼ ð10 À4 , .…”

“…11 We are not aware of other simulation approaches in the context of interacting particles that use the measurement data for direct calculation of the forces. 12 In fact each of these parameters was chosen merely to reach a heuristic level of sufficiency. For example, we let Δt 0 /Δt = 2 −5 because Δt 0 / Δt = 2 −6 did not improve simulation accuracy, and we chose K = 32 to provide a sufficiently large neighbourhood of cells for model replacement.…”

“…We solve the linear system (3.4) for coefficients boldw^false(ifalse) by approximately solving the following constrained sparse regression problem:boldw^false(ifalse)=arg minw s.t. Cw≤d⁡falsefalse{falsefalse‖boldGfalse(ifalse)w−boldbfalse(ifalse)‖22+λ2falsefalse‖w‖0falsefalse}.The linear inequality constraint C w ≤ d encodes the constraints listed in (2.2), (2.3) and (2.4) on the forces on f a−r , f align and f drag , and λ is the sparsity threshold. We employ the modified sequential thresholding algorithm from [12,63], with least-squares iterations replaced by solving the associated linearly constrained quadratic program. 5…”

“…While the list of hyperparameters is quite long (rows of table 5 with entries in last column, 18 in total), many can be taken sufficiently large ( N , K , S max , n gmm ), sufficiently small (Δ t ′, λ log , N ′ min ), or sufficiently dense (λ), depending purely on available computational resources. 12 The family of test functions Φ given by (3.5) has been demonstrated to work in a wide range of scenarios [12,62,63], and so may be taken as a fixed hyperparameter with minor tuning of the test function hyperparameters false(τ,τ^false), although it is possible that new application areas will require a different choice of test function class.…”