Sequential data assimilation for PDEs using shape-morphing solutions

TL;DR

DA-SMS provides a predictor–corrector framework for assimilating observations into shape-morphing solutions (SMS) of time-dependent PDEs, using Newton-like corrections to steer the SMS parameters toward the true state. The discrete-time scheme yields convergence under suitable data richness and regularization, while a continuous-time variant addresses high-frequency observations. Numerical tests on the nonlinear Schrödinger equation, Kuramoto–Sivashinsky equation, and 2D advection–diffusion demonstrate that DA-SMS achieves higher fidelity and longer predictive horizons than SMS alone, even with sparse or noisy data. A boundary-condition encoding technique within the SMS architecture enhances practicality for complex PDEs, highlighting DA-SMS as a scalable, data-informed alternative to traditional adjoint-based or Kalman-type methods.

Abstract

Shape-morphing solutions (also known as evolutional deep neural networks, reduced-order nonlinear solutions, and neural Galerkin schemes) are a new class of methods for approximating the solution of time-dependent partial differential equations (PDEs). Here, we introduce a sequential data assimilation method for incorporating observational data in a shape-morphing solution (SMS). Our method takes the form of a predictor-corrector scheme, where the observations are used to correct the SMS parameters using Newton-like iterations. Between observation points, the SMS equations (a set of ordinary differential equations) are used to evolve the solution forward in time. We prove that, under certain conditions, the data assimilated SMS (DA-SMS) converges uniformly towards the true state of the system. We demonstrate the efficacy of DA-SMS on three examples: the nonlinear Schrodinger equation, the Kuramoto-Sivashinsky equation, and a two-dimensional advection-diffusion equation. Our numerical results suggest that DA-SMS converges with relatively sparse observations and a single iteration of the Newton-like method.

Figures (6)

• Figure 1: Schematic illustration of the DA-SMS algorithm.
• Figure 2: Here we compare the modulus of the envelope $u(x,t)$ over the time interval $[0,150]$ using the initial parameters $A_0 = 0.2, \quad L_0 = 20, \quad V_0 = 0, \quad \phi_0 = 0$. We show (a.) DNS, (b.) SMS, (c.) DA-SMS. In (d.) we compare the envelopes of the three solutions at $x=0$, where DNS corresponds to the dashed red line, DA-SMS is the solid grey line, and SMS is the dot dashed blue line. The shaded red region denotes one standard deviation (5%) away from the DNS solution.
• Figure 3: Comparisons of the state of the system over the entire spatial domain $\Omega = [-11,11]$ and the entire duration of the simulation $t\in [0,100]$. The column on the left shows the results for: (a) DNS, (b) DA-SMS with clean (0% noise) data, (c) DA-SMS with 5% noise in the data, and (d) SMS with no data assimilation. In the column on the right we present the corresponding error defined by: $u_{DNS}(x,t) - \hat{u}(x,t)$ for (e) DA-SMS with clean data, (f) DA-SMS with noisy data, (g) SMS without data assimilation.
• Figure 4: Comparison of the relative error (RE) for DA-SMS with different parameter values. The errors are reported at the end of the data assimilation time window ($t=30$). The first column in each panel shows the SMS error, without data assimilation, for comparison. (a) Varying the Tikhonov regularization parameter $\tilde{\gamma}$. (b) Different sampling rate $\Delta t$. (c) Different number of sensors $r$. (d) Perturbing the sensor locations.
• Figure 5: Schematic illustration of the boundary conditions for the advection-diffusion equation in dimensionless variables. The domain size is $[0,4]\times [0,1]$. The white dots indicate the sensor locations for DA-SMS.

Theorems & Definitions (5)

• Lemma 1
• proof
• Theorem 1
• proof
• Remark 1