Physics-guided curriculum learning for the identification of reaction-diffusion dynamics from partial observations

TL;DR

This work tackles the ill-posed problem of identifying parameters and reconstructing hidden states in partially observed reaction-diffusion systems. It introduces CLIP, a physics-guided curriculum learning framework that integrates PINNs with a three-stage training schedule and an anchored widening transfer strategy, complemented by residual-based adaptive sampling. CLIP delivers superior accuracy and robustness across canonical RD benchmarks and a high-dimensional Min-system, with ablation studies and loss-landscape analyses confirming improved trainability and convergence. The approach offers a practical pathway to reliable RD dynamics inference from partial data and suggests a broader strategy for multi-physics identification with stage-wise learning.

Abstract

Reaction-diffusion (RD) systems provide fundamental models for understanding self-organized spatiotemporal patterns across diverse natural and engineered settings, but reliable parameter estimation remains challenging, particularly when observations are sparse, noisy, and restricted to a subset of state variables. Based on physics-informed neural networks (PINNs), a physics-guided Curriculum Learning Identification via PINNs (CLIP) method is introduced in this work, for joint parameter inference and hidden state reconstruction. Leveraging the physical separability of RD systems, the CLIP training progresses from reaction-dominated regimes to full spatiotemporal dynamics using curriculum learning and an anchored widening transfer strategy. Across three canonical reaction-diffusion benchmarks, CLIP achieves more accurate and robust identification than baseline methods. Furthermore, the CLIP framework is successfully applied to infer the dynamics of Min system in bacteria, where only membrane bound species are observed and key kinetic rates span multiple orders of magnitude. Moreover, ablation experiments and loss landscape analyses provide mechanistic evidence that the curriculum stages and anchored transfer enhance trainability and convergence.

Figures (4)

• Figure 1: The proposed CLIP framework.a, Representative spatiotemporal pattern types generated by the two-dimensional Gray–Scott reaction–diffusion model. b, Overview of the physics-induced Curriculum Learning Identification via PINNs (CLIP) training scheduler illustrated using a two-variable reaction–diffusion system. A physics-guided curriculum design stratifies the training data and associated physics constraints into a sequence of progressively more challenging subtasks. Curriculum regions are defined using $\Delta u$, the Laplacian of the observed field. The Gray–Scott system is shown as an example using a snapshot of the $u$ field at time step $t_s=600$. CLIP proceeds through three stages (primary, middle and fine-tuning stages), which successively identify reaction parameters, estimate diffusion coefficients, and jointly refine parameters and state reconstructions. c, Architecture and training strategy of the multi-task neural network. A shared spatiotemporal encoder takes coordinates $(\mathbf{x},t)$ as input, followed by one task-specific branch for each state variable. Observable states are directly constrained by data losses, while unobserved states are reconstructed jointly through physics-based coupling. Training is conducted in multiple stages, during which different subsets of data and physics constraints are progressively activated. The losses $\mathcal{L}_{\mathrm{data}}$, $\mathcal{L}_{\mathrm{pde}}$, and $\mathcal{L}_{\mathrm{ic}}$ denote data mismatch, PDE residual, and initial-condition consistency, respectively. $N_k$ denotes the number of spatiotemporal training points $\{(\mathbf{x}_k,t_k)\}_{k=1}^{N_k}$. The data losses in primary stage are evaluated on reaction-dominated samples selected by a spatiotemporal mask. Detailed definitions of the losses and mask construction are provided in Methods.
• Figure 2: Performance of CLIP framework on three reaction–diffusion systems. In the governing equations, variables shown in blue are unobserved, and parameters shown in red are inferred. a–c, Results on the $\lambda$–$\omega$ system; d–f, Gray–Scott; g–i, Lotka–Volterra. a,d,g, Reconstructed unobservable fields with comparisons to baseline methods. b,e,h, Single-point time series at the domain center. c,f,i, Relative errors of identified parameters from ablation on curriculum learning and anchored widening transfer learning.
• Figure 3: Performance of CLIP framework on the Min proteins dynamics. a. The governing equations of the Min system, where blue denotes unobservable state variables and red indicates parameters to be identified. The table reports the reference values and the identified parameters on clean data. b. Reconstructed unobservable fields with comparisons to the baseline PINN. c. Single-point time series at the domain center. d. Relative errors of identified parameters from ablation on curriculum learning and anchored widening transfer learning.
• Figure 4: Visualization of the loss landscape. Two-dimensional visualization of the loss surface and projected learning trajectories obtained by PCA. Green, blue, and red markers denote the training start, the converged solutions, and optimization trajectory, respectively. a, The loss surface of the baseline PINN results. b-d, The loss surface of the three successive stages of CLIP training.