NAS-PINN: Neural architecture search-guided physics-informed neural network for solving PDEs

• Traditional mesh-based numerical methods can be precise and mathematically rigorous, but their computational cost grows exponentially with PDE dimension.Source: §1 · p. 1
• Neural-operator approaches can handle equations with shared form but different parameters after training, yet require substantial numerical data and remain data-driven.The introduction also notes costs from collecting training data and optimizing network parameters.Source: §1 · pp. 1–2
• PINNs embed physical information through loss functions and use automatic differentiation for partial derivatives; related work explores domains, boundary conditions, gradients, differentiation, and sampling [17].Source: §1 · p. 2
• PINN architectures have largely followed experience-based routines of four to six hidden layers with equal neuron counts, while prior architecture-performance studies are described as fragmentary and time-consuming [21].Source: §1 · p. 2
• Traditional neural architecture search evaluates discrete candidate networks, while differentiable NAS relaxes the search space for gradient-based optimization.The introduction also discusses efficiency approaches including progressive pruning and parameter sharing.Source: §1 · p. 2
• NAS-PINN incorporates architecture search into PINNs, uses masks to add tensors of different shapes, and summarizes effective architectures across PDE experiments.The stated aim is to search automatically for an architecture suited to a given PDE.Source: §1 · p. 2

• The stated PDE form combines a time derivative and spatial differential operator with a source term, alongside boundary and initial conditions.Source: §2 · p. 3
• The formulation covers a computational domain and its boundary over a time interval, and NAS-PINN is applied to PDEs of this form.Source: §2 · p. 3
• The unknown is the latent solution; F is the spatial differential operator, while B and I calculate boundary and initial values.Source: §2 · p. 3

3.1 Physics-Informed Neural Network (PINN)

• PINN takes spatial and temporal collocation coordinates as input and predicts a solution, with a loss designed to enforce the PDE.The loss can include domain, boundary, and initial-condition terms, weighted and evaluated over their respective collocation points.Source: §3.1 · pp. 3–4

3.2 Differentiable NAS

• Differentiable NAS relaxes candidate operations into a continuous search space optimized by gradient methods, targeting DNN layer counts and neurons per layer in PINNs.After convergence, the candidate operation with the highest learned weight can be selected to obtain a discrete architecture.Source: §3.2 · p. 4

3.3 Masks

• Zero-padding and one-zero masks enable neuron-count choices despite tensor shape constraints, while identity transformations let the search skip layers.Shared weights reduce optional hidden layers to one; architecture weights determine neuron counts and whether layers are retained.Source: §3.3 · pp. 4–5, Figure 2 · p. 5

3.4 NAS-PINN

• NAS-PINN uses bi-level optimization: the inner loop optimizes network weights and bias, while the outer loop searches architecture parameters.Alternate optimization is used, and the outer-loop loss can use a rather small number of data points. After training, architecture weights determine a discrete network; near-equal skip and retain weights can instead yield a mixed model.Source: §3.4 · pp. 5–6

4.1 Poisson equation

• For square-domain Poisson, NAS-PINN found the best architecture in the search space, while a shallow, wide network outperformed deeper ones.The searched architecture had the smallest L2 and maximum errors; the table reports L2 errors for the candidate architectures.Source: §4.1 · p. 7, Table 1 · p. 7, Figure 4 · p. 8

4.2 Poisson equation in irregular computational domains

• On irregular Poisson domains, NAS-PINN adapted well, and residual connections were favored for complex problems, often in middle and later layers.Dumpy outperformed Giant and Slender on circular and L-shaped domains, while NAS-PINN and Giant prevailed on the flower-shaped domain.Source: §4.2 · pp. 8, 11, Table 2 · p. 9

4.3 Burgers equation

• For 1-D Burgers, NAS-PINN predictions were more accurate than the references, especially at υ = 0.1, and generally used three or four hidden layers.The comparisons indicate that more layers may matter more than more neurons for Burgers, while deeper networks do not guarantee better results.Source: §4.3 · p. 11, Table 3 · p. 11, Figure 8 · p. 12

4.4 Advection equation

• For Advection, NAS-PINN architectures achieved the best results, with shallower networks selected as β increased.The reference architecture rankings also varied with β, and larger β values corresponded to higher L2 error.Source: §4.4 · p. 13, Table 4 · p. 13, Figure 10 · p. 14

4.5 2-D Burgers equation

• NAS-PINN achieved the minimum L2 error on 2-D Burgers, improving accuracy by an order of magnitude over four reference architectures.Giant had accuracy close to NAS-PINN but substantially more parameters; more layers helped up to a problem-specific optimum, while too many could hurt.Source: §4.5 · p. 15, Table 5 · p. 15

5 CONCLUSIONS

• NAS-PINN relaxes architecture search into a continuous, bi-level optimization that selects hidden-layer counts and per-layer neuron counts using mixed operations and masks.Source: §5 · p. 15
• More hidden layers do not guarantee better performance: shallower, wider networks can prevail for Poisson and Advection, while Burgers favors more layers.Source: §5 · p. 15
• Effective architectures vary across PDEs, and different neuron counts across layers are preferable to uniform-width hidden layers.Source: §5 · pp. 15, 17
• This PDE-specific diversity complicates manual design, while NAS-PINN can help characterize efficient architectures and guide future designs.Source: §5 · p. 17
• NAS-PINN has so far focused on DNNs; extending its search to CNNs and choosing between DNNs and CNNs remain future directions.Source: §5 · p. 17