DDC-PINNs: A Predictor-Corrector Approach Based on Neural Network-Driven Domain Decomposition and Classical ODE Solvers for Time-Dependent PDEs

TL;DR

DDC-PINNs combines domain decomposition, causality constraints, and classical ODE solvers to improve the accuracy and efficiency of neural network solutions for time-dependent PDEs.

Contribution

It introduces a novel predictor-corrector framework that enhances PINNs with domain decomposition and classical ODE solvers for better handling of time-dependent PDEs.

Findings

Improved accuracy in solving time-dependent PDEs.

Effective incorporation of causality constraints.

Numerical experiments demonstrate superior performance.

Abstract

When solving time-dependent partial differential equations(PDEs), traditional physics-informed neural networks (PINNs) have inherent limitations: due to the lack of temporal causality, the network is forced to learn the later-time control equations while fully capturing the initial conditions, resulting in the continuous accumulation of errors during the integration process. Meanwhile, the limited expressivity of a single network hinders its ability to capture diverse physical behaviors across multiple subdomains. To address these issues, we propose a domain-decomposition-based causal PINNs (DDC-PINNs) framework. This framework enhances spatial representation through domain decomposition and employs a causal strategy to constrain the temporal learning sequence, thereby improving the accuracy and generalization ability of solutions to time-varying problems. Within this framework, an approximate solution is first obtained through PINNs with domain decomposition. Subsequently, the time derivative term in the PDE is retained, while other solution-dependent terms are replaced with this approximate solution, thereby simplifying the original PDEs into ordinary differential equations (ODEs). Finally, classical numerical methods for solving ODEs are employed to obtain the time-dependent solution. DDC-PINNs not only preserve the inherent computational efficiency and flexibility of PINNs but also effectively incorporate causality when solving time-dependent PDEs. Numerical experiments verify the effectiveness of the proposed method.

Tables

Table 1. Table 1 : Comparison of relative errors of XPINNs, IDPINNs and DDC-PINNs Methods for Example 5.1

Methods	PINNs	XPINNs	IDPINNs	DDC-PINNs
Relative $L^2$ errors	$1.57\times {10}^{-2}$	$1.64\times {10}^{-1}$	$7.85\times {10}^{-2}$	$7.11\times {\mathrm{𝟏𝟎}}^{-\mathrm{𝟑}}$

Table 2. Table 2 : Relative errors comparison: Neural Galerkin, CEENs, and DDC-PINNs for Example 5.1

Methods	Neural Galerkin	CEENs	DDC-PINNs
Relative $L^2$ errors	$1.22\times {10}^{-2}$	$3.06\times {10}^{-2}$	$7.11\times {\mathrm{𝟏𝟎}}^{-\mathrm{𝟑}}$
Time	1487.5s	36172.8s	1899.5s

Table 3. Table 3 : Evaluation of relative errors for XPINNs, IDPINNs and DDC-PINNs Methods for Example 5.2 .

Methods	PINNs	XPINNs	IDPINNs	DDC-PINNs
Relative $L^2$ errors	$1.07\times {10}^{-2}$	$1.96\times {10}^{-1}$	$3.72\times {10}^{-3}$	$3.31\times {\mathrm{𝟏𝟎}}^{-\mathrm{𝟑}}$

Table 4. Table 4 : Relative errors comparison: Neural Galerkin, CEENs, and DDC-PINNs for Example 5.2

Methods	Neural Galerkin	CEENs	DDC-PINNs
Relative $L^2$ errors	$5.40\times {10}^{-3}$	$3.33\times {10}^{-2}$	$3.31\times {\mathrm{𝟏𝟎}}^{-\mathrm{𝟑}}$
Time	880.7s	6526.5s	1195.2s

Table 5. Table 5 : The relative errors of XPINNs, IDPINNs, and DDC-PINNs are evaluated for Example 5.3 .

Methods	PINNs	XPINNs	IDPINNs	DDC-PINNs
Relative $L^2$ errors	$1.08\times {10}^{-2}$	$1.17\times {10}^{-2}$	$4.11\times {10}^{-3}$	$2.09\times {\mathrm{𝟏𝟎}}^{-\mathrm{𝟑}}$

Table 6. Table 6 : Relative errors of Neural Galerkin, CEENs, and DDC-PINNs for Example 5.3 .

Methods	Neural Galerkin	CEENs	DDC-PINNs
Relative $L^2$ errors	$4.60\times {10}^{-3}$	$3.23\times {10}^{-2}$	$2.09\times {\mathrm{𝟏𝟎}}^{-\mathrm{𝟑}}$
Time	189.8s	23234.1s	984.4s

Table 7. Table 7 : Relative errors of XPINNs, IDPINNs, and DDC-PINNs for Example 5.4 .

Methods	PINNs	XPINNs	IDPINNs	DDC-PINNs
Relative $L^2$ errors	$4.83\times {10}^{-2}$	$7.14\times {10}^{-2}$	$2.56\times {10}^{-2}$	$4.48\times {\mathrm{𝟏𝟎}}^{-\mathrm{𝟑}}$

Table 8. Table 8 : The relative errors of Neural Galerkin, CEENs, and DDC-PINNs are compared for Example 5.4 .

Methods	Neural Galerkin	CEENs	DDC-PINNs
Relative $L^2$ errors	$7.08\times {\mathrm{𝟏𝟎}}^{-\mathrm{𝟒}}$	$9.85\times {10}^{-2}$	$4.48\times {10}^{-3}$
Time	5659.6s	462180.0s	2431.1s

Equations

$$u_t(\mathbf{x},t)+\mathcal{N}[u,\mathbf{x},t]=0,\mathbf{x}\in \Omega ,t\in [0,T],$$

$$u(\mathbf{x},0)=h(\mathbf{x}),\mathbf{x}\in \Omega ,$$

$$\mathcal{B}[u(\bm{x},t)]=b(\bm{x},t),\quad\bm{x}\in\partial\Omega,\,t\in[0,T]{\color[rgb]{0,0,0}.}$$

$$u(\mathbf{x},t)\approx \arg \underset{\hat{u}\in F}{\min }J(\theta ),$$

$$F:=\{\hat{u}(\cdot,\cdot,\theta):\theta\in\mathbb{R}^{P}\}\subset C(\bar{\Omega}\times[0,T]){\color[rgb]{0,0,0}.}$$

$$J(\theta )={\lambda }_1{\mathcal{L}}_r(\theta )+{\lambda }_2{\mathcal{L}}_b(\theta )+{\lambda }_3{\mathcal{L}}_0(\theta ),$$

$${\mathcal{L}}_r(\theta )$$

$${\mathcal{L}}_b(\theta )$$

$${\mathcal{L}}_0(\theta )$$

$${\mathcal{L}}_{IF}(\theta )=\sum _{ij}\frac{1}{N_{I{F}_{ij}}}\sum _{q=1}^{N_{I{F}_{ij}}}|{\hat{u}}_t^i({\mathbf{x}}_q^{ij},t_q^{ij},\theta )+\mathcal{N}[{\hat{u}}^i,{\mathbf{x}}_q^{ij},t_q^{ij}]-{\hat{u}}_t^j({\mathbf{x}}_q^{ij},t_q^{ij},\theta )-\mathcal{N}[{\hat{u}}^j,{\mathbf{x}}_q^{ij},t_q^{ij}]{|}^2,$$

$${\mathcal{L}}_I(\theta )=\sum _{ij}\frac{1}{N_{I{F}_{ij}}}\sum _{q=1}^{N_{I{F}_{ij}}}|{\hat{u}}^i({\mathbf{x}}_q^{ij},t_q^{ij},\theta )-\frac{1}{2}[{\hat{u}}^i({\mathbf{x}}_q^{ij},t_q^{ij},\theta )+{\hat{u}}^j({\mathbf{x}}_q^{ij},t_q^{ij},\theta )]{|}^2,$$

$$J(\theta {)}_{XP}={\lambda }_1{\mathcal{L}}_r(\theta )+{\lambda }_2{\mathcal{L}}_b(\theta )+{\lambda }_3{\mathcal{L}}_0(\theta )+{\lambda }_4{\mathcal{L}}_{IF}(\theta )+{\lambda }_5{\mathcal{L}}_I(\theta ),$$

$$J(\theta {)}_{X{P}_{total}}=J(\theta {)}_{XP}+{\lambda }_6{\mathcal{L}}_{\nabla }\mathfrak{F}(\theta )+{\lambda }_7{\mathcal{L}}_{\nabla }{u}_c(\theta )+{\lambda }_8{\mathcal{L}}_{\nabla }{\mathfrak{F}}_c(\theta )+{\lambda }_9\mathcal{L}\mathfrak{F}(\theta ),$$

$${\mathcal{L}}_{\nabla }\mathfrak{F}(\theta )=\sum _{ij}\sum _{k=i,j}\frac{1}{N_{I{F}_{ij}}}\sum _{q=1}^{N_{I{F}_{ij}}}|\nabla ({\hat{u}}_t^k({\mathbf{x}}_q^{ij},t_q^{ij},\theta )+\mathcal{N}[{\hat{u}}^k,{\mathbf{x}}_q^{ij},t_q^{ij}]){|}^2,$$

$${\mathcal{L}}_{\nabla }{u}_c(\theta )=\sum _{ij}\frac{1}{N_{I{F}_{ij}}}\sum _{q=1}^{N_{I{F}_{ij}}}|\nabla {\hat{u}}^i({\mathbf{x}}_q^{ij},t_q^{ij},\theta )-\nabla {\hat{u}}^j({\mathbf{x}}_q^{ij},t_q^{ij},\theta ){|}^2,$$

$${\mathcal{L}}_{\nabla }{\mathfrak{F}}_c(\theta )=\sum _{ij}\frac{1}{N_{I{F}_{ij}}}\sum _{q=1}^{N_{I{F}_{ij}}}|\nabla ({\hat{u}}_t^i({\mathbf{x}}_q^{ij},t_q^{ij},\theta )+\mathcal{N}[{\hat{u}}^i,{\mathbf{x}}_q^{ij},t_q^{ij}])$$

$$-\nabla ({\hat{u}}_t^j({\mathbf{x}}_q^{ij},t_q^{ij},\theta )+\mathcal{N}[{\hat{u}}^j,{\mathbf{x}}_q^{ij},t_q^{ij}]){|}^2,$$

$${\mathcal{L}}_{\mathfrak{F}}(\theta )=\sum _{ij}\sum _{k=i,j}\frac{1}{N_{I{F}_{ij}}}\sum _{q=1}^{N_{I{F}_{ij}}}|{\hat{u}}_t^k({\mathbf{x}}_q^{ij},t_q^{ij},\theta )+\mathcal{N}[{\hat{u}}^k,{\mathbf{x}}_q^{ij},t_q^{ij}]{|}^2.$$

$$\mathcal{N}[u,\mathbf{x},t]\approx \mathcal{N}[\hat{u},\mathbf{x},t].$$

$$u_t(\mathbf{x},t)\approx -\mathcal{N}[\hat{u},\mathbf{x},t],\mathbf{x}\in \Omega ,t\in [0,T],$$

$$u(\mathbf{x},0)=h(\mathbf{x}),\mathbf{x}\in \Omega .$$

$${\mathcal{E}}_{\mathrm{total}}=E({\mathcal{E}}_{\mathrm{app}},{\mathcal{E}}_{\mathrm{gen}},{\mathcal{E}}_{\mathrm{opt}})$$

$$u_t(\mathbf{x},t)+\mathcal{N}[u,\mathbf{x},t]=0,u(\mathbf{x},0)=h(\mathbf{x}),$$

$${\varphi }_t(\mathbf{x},t)=-\mathcal{N}[\hat{u},\mathbf{x},t],\varphi (\mathbf{x},0)=h(\mathbf{x}),$$

$$e_t(\mathbf{x},t)=-\mathcal{N}[\hat{u},\mathbf{x},t]+\mathcal{N}[u,\mathbf{x},t]=\mathcal{N}[u,\mathbf{x},t]-\mathcal{N}[\hat{u},\mathbf{x},t].$$

$$e_t(\mathbf{x},t)=\mathcal{N}[\hat{u}+\delta ,\mathbf{x},t]-\mathcal{N}[\hat{u},\mathbf{x},t].$$

$${\Vert \mathcal{N}[w_1(\cdot ,t),\mathbf{x},t]-\mathcal{N}[w_2(\cdot ,t),\mathbf{x},t]\Vert }_Y\le L{\Vert w_1(\cdot ,t)-w_2(\cdot ,t)\Vert }_X.$$

$$e(\mathbf{x},s)={\int }_0^s(\mathcal{N}[\hat{u}+\delta ,\mathbf{x},t]-\mathcal{N}[\hat{u},\mathbf{x},t])dt.$$

$${\Vert e(\mathbf{x},s)\Vert }_Y={\Vert {\int }_0^s\mathcal{N}[\hat{u}+\delta ,\mathbf{x},t]-\mathcal{N}[\hat{u},\mathbf{x},t]dt\Vert }_Y\le {\int }_0^s{\Vert \mathcal{N}[\hat{u}+\delta ,\mathbf{x},t]-\mathcal{N}[\hat{u},\mathbf{x},t]\Vert }_{Y}dt.$$

$${\Vert e(\mathbf{x},s)\Vert }_Y\le L{\int }_0^s{\Vert \delta (\mathbf{x},t)\Vert }_{X}dt.$$

$$\text{L2-error}=\frac{\sqrt{\sum _{j=1}^N|\hat{u}({\mathbf{x}}_j,t_j,\cdot )-u_{\text{ref}}({\mathbf{x}}_j,t_j){|}^2}}{\sqrt{\sum _{j=1}^N|u_{\text{ref}}({\mathbf{x}}_j,t_j){|}^2}},$$

$$u_t=u_{xx}+e^{-t}({\pi }^2\sin (\pi x)-\sin (\pi x)),x\in [-1,1],t\in [0,1],$$

$$u(-1,t)=u(1,t)=0,t\in [0,1],$$

$$u(x,0)=\sin (\pi x),x\in [-1,1].$$

$$u_t=-u{u}_x,x\in [-1,1],t\in [0,1],$$

$$u(-1,t)=u(1,t),t\in [0,1],$$

Full text

1 Introduction

The advent of deep learning has revolutionized computational methods in various domains, including speech processing, natural language understanding, and computer vision JGu , LECUN , KRIZHEVSKY , HINTON . Within this technological landscape, Raissi et al. proposed PINNs Raissi , initiating a new paradigm for applying deep learning to solve forward and inverse problems in PDEs, and providing a potent alternative to conventional numerical PDEs solvers. Integrating PDEs constraints into the loss function via automatic differentiation, PINNs solve PDEs. The advantages of PINNs are mesh-free and structural simplicity. Thus, PINNs can be used in various complex PDEs, such as fractional-order GPang , LingGuo , integro-differentialLLu , LeiYuan , and stochastic PDEs DZhang , DZhang1 . Recent research has not only improved the accuracy of PINNs in solving PDEs Jeremy , Sifan , Zixue , Sokratis , SWang , LeviD , Yanjie , YiqiGu , Chenxi , but has also extended their applications to other disciplines CAIS , MENGX , PANGG , YChen .

Despite its many advantages in solving PDEs, PINNs also have certain limitations. First, PINNs struggle to explicitly model temporal causality, which leads them to favor solutions that satisfy the governing equations at later time steps before the initial conditions have been adequately learned, thereby introducing cumulative errors. Second, the training process is computationally expensive, particularly for complex or high-dimensional problems. A single network often finds it difficult to simultaneously capture the differences in physical behavior across multiple subregions, and when the solution exhibits sharp variations in space or time, the expressive power of a global model tends to be insufficient.

To address these challenges, researchers have proposed various methods. For instance, Du et al. introduced the Evolutionary Deep Neural Network (EDNN), a novel framework for solving PDEs YDu . This algorithm treats the network parameters as variables that evolve over time and updates them dynamically, while embedding constraints such as boundary conditions directly into the network architecture. This design enables EDNN to accurately capture the causality in the solutions of PDEs. Bruna et al. combined time-parametrized network weights with the Dirac-Frenkel variational principle. Based on the strong form of the equation, they transformed the PDEs into an ODEs, thereby enabling its solution via classical numerical differentiation methods, which ensures that the solution possesses causal properties JBruna , JBerman , YuxiaoWen , Zhang , HuZiqing , Schwerdtner . Building on this, Yang et al. first transformed the strong form of the equation into a weak form and then combined it with the Dirac-Frenkel variational principle; this method also captured the causality of the solutions XYang ; the accuracy of the neural Galerkin method and the weak neural Galerkin method varies depending on the specific case, but their overall accuracy is comparable. On the other hand, Jung et al. proposed Causal Evolutionary Reinforcement Networks (CEENs)JungJ . In this method, time integration is applied to both sides of the time-dependent PDEs. By ingeniously employing the trapezoidal rule for numerical time integration, they constructed an equation that describes the causal relationship of the solution. Subsequently, they performed training and optimization at each time step, ultimately obtaining the causality of the solution to this equation.

Jagtap et al. proposed conservative PINNs (cPINNs), which enforce interface continuity conditions across partitioned domains Jagtap , while XPINNs extend domain decomposition applicability to broader range of PDEs ADJagtap . Recently, Dolean et al. presented a multi-level domain-decomposition-based PINNs framework DoleanV . Compared to FBPINNs Moseley , it uses a multi-level solution representation. Different from XPINNs, subdomains of FBPINNs can be overlapped, that avoids the introduction of extra interface loss terms. This method is mainly used in high-frequency problems. Kim et al. introduced Initialization-enhanced PINNs (IDPINNs), which optimizes interface continuity in XPINNs and contains a strategy for initializing training data Chenhao . Hu et al. presented a non-overlapping Schwarz-type domain decomposition method for physics and equality constrained artificial neural networks QifengHu . Based on domain decomposition, it transforms PDEs problems into optimization ones with equality constraints. Its loss function, built by interface information, could be solved by the adaptive augmented Lagrangian method. Shang et al. proposed a method that integrates subdomain-decomposition-based stochastic neural networks with overlapping Schwarz preconditioners for PDEs solving ShangY . This method combines subdomain-decomposition which based on stochastic neural networks with overlapping Schwarz preconditioners. Subsequently, more researchers have explored PINNs methods based on domain decomposition techniques. These developments collectively enhance computational efficiency and theoretical basis for complex system modeling Ye , ZHu , Hadden .

To enhance the flexibility of spatial domain processing and describe the causal relationships among the solutions, we introduce DDC-PINNs, a novel framework that combines neural networks with domain decomposition and classical numerical solutions for solving time-dependent PDEs. The method proceeds through two phases: (1) Initial solution approximation given by DDPINNs with enhanced interface treatment; (2) Based on the initial approximation, we retain the time derivative term in the PDEs while replacing all other solution-dependent terms with the current approximate solution, thereby transforming the original problem into an ODEs system, which is then solved using classical numerical methods for temporal evolution. This approach provides a novel framework for solving time-dependent PDEs that retains the efficiency and flexibility of PINNs and improves the capability of dynamically solving time-dependent PDEs.

The advantages of our method are as follows:

Causality of the numerical solution: Compared to PINNs (Raissi ) and other existing methods (e.g., Jagtap , ADJagtap , DoleanV , Moseley , Chenhao , QifengHu , ShangY ), the numerical solutions obtained by solving PDEs using DDC-PINNs capture the causal relationships within the solutions.

• 2.

Reduced Training Complexity: DDC-PINNs reduce the computational cost of long-term simulations compared to fully-connected approaches such as CEENs.

• 3.

Enhanced Scalability & Robustness: Compared to XPINNs (ADJagtap ) and IDPINNs (Chenhao ), an optimized domain decomposition method is employed. Furthermore, subsequent numerical experiments verify that DDC-PINNs achieve higher computational accuracy than the existing time-varying causal solver CEENs, and also outperform the neural Galerkin method on certain problems.

• 4.

Hybrid Efficiency & Flexibility: DDC-PINNs combine the advantages of neural networks with the efficiency and reliability of classical numerical integration. DDC-PINNs offer a flexible framework for time-dependent problems and retain the versatility of PINNs.

This paper is organized as follows: In section 2, PINNs for solving PDEs are reviewed. In section 3, the novel framework DDC-PINNs is introduced. In Section 4, an error analysis of DDC-PINN is presented. In Section 5, the effectiveness of our method is demonstrated through a series of numerical experiments; and our conclusions are presented in Section 6.

2 Review of PINNs

We review the application of PINNs in solving PDEs in this section. We begin by formulating the general time-dependent PDEs system

[TABLE]

governed by initial conditions

[TABLE]

and boundary conditions

[TABLE]

Here, $\Omega$ is a bounded region in $\mathbb{R}^{d}$($d$ is the dimension of the spatial region), $\mathcal{N}[u,\bm{x},t]$ denotes a linear or nonlinear partial differential operator. The operator $\mathcal{B}[u(\bm{x},t)]$ represents the behavior of the solution at the boundary $\partial\Omega\times[0,T]$, and $h(\bm{x})$ represents the initial condition at $t=0$.

In the PINNs framework, we employ fully connected neural networks to estimate solutions for PDEs. The output of the neural network $\hat{u}(\bm{x},t,\theta)$ serves as an approximate solution, where $\theta$ denotes the parameters of the network. By optimizing $\theta$, it aims to improve the accuracy of $\hat{u}$ in approximating the true solution. This optimization is formulated as:

[TABLE]

where

[TABLE]

Here, $P$ denotes the number of weights and bias terms of the neural networks, and the loss function $J(\theta)$ comprises three key components:

[TABLE]

where $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ are weighting factors that balance the contributions of the PDEs residual, boundary conditions, and initial conditions, respectively. The components of the loss function of PINNs are defined as follows:

[TABLE]

where, $(\bm{x}_{r}^{j},t_{r}^{j})\in\Omega\times[0,T]$, $(\bm{x}_{b}^{j},t_{b}^{j})\in\partial\Omega\times[0,T]$, $\bm{x}_{0}^{j}\in\Omega$. $N_{r}$, $N_{b}$ and $N_{0}$ denote the numbers of collocation points for the PDE residual, boundary conditions and initial conditions, respectively.

3 DDC-PINNs for solving temporal PDEs

In this section, we present DDC-PINNs, a novel framework for solving temporal PDEs. First, we introduce DDPINNs as its foundational component.

3.1  DDPINNs for solving PDEs

Domain decomposition is based on the principle of dividing a large domain into smaller domains, with each subdomain is trained by a separate neural network. Aiming to improve the accuracy and smoothness of the XPINNs solution at the interface, we add residual, gradient residual, and continuity terms for both residual gradients and approximate solution gradients at the interface to the total loss function.

Let $\Omega_{1}\cup\cdots\cup\Omega_{i}\cup\cdots\cup\Omega_{j}\cup\cdots\cup\Omega_{N}=\Omega$ $(1\leq i<j\leq N)$ . If $\Omega_{i}$, $\Omega_{j}$ are adjacent, then $\Omega_{i}\cap\Omega_{j}=\partial\Omega_{ij}$; otherwise $\Omega_{i}\cap\Omega_{j}=\emptyset$. The training points used for residuals in subdomain $\Omega_{i}$ are denoted by $\{(\bm{x}_{q}^{i},t_{q}^{i})\}_{q=1}^{N_{r_{i}}}$. ${N_{r_{i}}}$ denotes the total number of training points on subdomain $\Omega_{i}$, where $(\bm{x}_{q}^{i},t_{q}^{i})$ $\in\Omega_{i}\times(0,T)$. If domain $\Omega_{i}$ and domain $\Omega_{j}$ are adjacent, $\{(\bm{x}_{q}^{ij},t_{q}^{ij})\}_{q=1}^{N_{IF_{ij}}}\in\partial\Omega_{ij}$. $N_{IF_{ij}}$ denotes the number of training points at interface $\partial\Omega_{ij}$. The loss function of XPINNs is defined in the work of Jagtap et al ADJagtap . as follows:

[TABLE]

where $\mathcal{L}_{IF}(\theta)$ and $\mathcal{L}_{I}(\theta)$ are used to ensure continuity at the interface.

However, this treatment of the interface part is rough and cannot guarantee the smoothness of the interface part Chenhao . To deal with the interface part, we propose DDPINNs based on IDPINNs. The loss function of DDPINNs is constructed as follows:

[TABLE]

where

[TABLE]

3.2  DDC-PINNs solving PDEs

When solving time-dependent PDEs using traditional PINNs, it is difficult to capture the causal relationships of the solutions. To overcome this limitation, we propose the DDC-PINNs method, which is built on a domain decomposition framework. This method retains the flexibility of DDPINNs in handling complex problems while enhancing the ability to characterize the causal relationships of the solutions.

Using the approximate solution $\hat{u}$ obtained by DDPINNs, we establish the approximation:

[TABLE]

By this approximation, the original problem c-(2.3) transforms into the following initial-value problem of an ODE system:

[TABLE]

The system (3.10)-(3.11) can be solved by classical numerical methods such as the Euler scheme or Runge-Kutta methods. This computational strategy is named as DDC-PINNs.

Figure 1 illustrates the DDC-PINNs framework for solving time-dependent PDEs, where $\hat{u}$ denotes the approximate solution derived from the DDC-PINNs, and $\tilde{u}$ denotes the solution obtained via classical numerical methods at discrete time steps, which reflects the causality of the solution.

4 Error analysis

In this subsection, we analyse the error associated with the proposed DDC-PINNs framework. This error depends primarily on a pair of key factors: the initial approximation accuracy provided by the DDPINNs, and the numerical error introduced by the subsequent ordinary differential equation integrator. Following the error decomposition introduced for XPINNs ADJagtap .

4.1  Error Sources in the DDPINNs Approximation

Let the computational domain be partitioned into $N_{sd}$ nonoverlapping subdomains $\{\Omega_{q}\}^{N_{sd}}_{q=1}$, In each subdomain a separate neural network $\hat{u}=\Sigma_{q}\hat{u}_{q}\mathbf{1}_{\Omega_{q}}$ is trained ($\mathbf{1}_{\Omega_{q}}$ is an indicator function). The total error between the exact solution $u$ and the DDPINNs approximation

The total error can be decomposed as

[TABLE]

where

$\mathcal{E}_{\mathrm{app}}$ is the approximation error, arising from the limited capacity of the neural network to represent the exact solution. It can be reduced by increasing the depth/width of the network or by using adaptive activation functions.

• 2.

$\mathcal{E}_{\mathrm{gen}}$ is the generalization error, caused by the finite number and distribution of collocation points (residual, boundary, and interface points). This error depends on the sampling strategy and can be mitigated by placing more points in regions with complex solution behaviour or near interfaces.

• 3.

$\mathcal{E}_{\mathrm{opt}}$ is the optimization error, originating from the non-convex nature of the loss landscape and the difficulty of reaching the global minimum. It is influenced by the choice of optimizer, learning rate, and the number of training iterations. $E$ is an abstract function that depends on $\mathcal{E}_{\mathrm{app}}$, $\mathcal{E}_{\mathrm{gen}}$ and $\mathcal{E}_{\mathrm{opt}}$.

4.2  Error Propagation in the Reduced ODEs System

Let $u$ be the exact solution of the original PDEs

[TABLE]

and let $\hat{u}(\bm{x},t)$ be the DDPINNs approximation. In the DDC-PINNs framework, we solve the initial value problem

[TABLE]

where $\phi(\bm{x},t)$ denotes the DDC-PINNs solution. Define the error $e(\bm{x},t)=\phi(\bm{x},t)-u(\bm{x},t)$. Using (4.3)-(4.2), we obtain

[TABLE]

Let $\delta(\bm{x},t)=u(\bm{x},t)-\hat{u}(\bm{x},t)$ be the DDPINNs approximation error. Then $u(\bm{x},t)=\hat{u}(\bm{x},t)+\delta(\bm{x},t)$ and the error equation becomes

[TABLE]

For any time $t$, assume that the exact solution $u(\cdot,t)$ and the DDPINNs approximation $\hat{u}(\cdot,t)$ belong to a bounded convex set $B\subset X$ (e.g., $B=\{w\in H^{1}(\Omega):\|w\|_{H^{1}}\leq M\}$). On this bounded set, the differential operator $\mathcal{N}$ is locally Lipschitz continuous with respect to $u$. More precisely, there exists a constant $L>0$ (depending on $B$ but independent of $t$) such that for all $t\in[0,T]$ and any two functions $w_{1}(\cdot,t),w_{2}(\cdot,t)\in B$ (with $\mathcal{N}[w_{1},\bm{x},t],\mathcal{N}[w_{2},\bm{x},t]\in Y$), we have

[TABLE]

This assumption is justified because the true solution $u$ is bounded in appropriate norms (due to well-posedness of the PDEs), and the DDPINNs approximation $\hat{u}$ is sufficiently close to $u$, hence also lies in $B$ for all $t\in[0,T]$. Here the norms $\left\|\cdot\right\|_{X}$ and $\left\|\cdot\right\|_{Y}$ depend only on the spatial variable $\mathbf{x}$; the time variable $t$ is treated as a parameter.

We integrate the error equation $e_{t}(\bm{x},t)=\mathcal{N}[\hat{u}+\delta,\bm{x},t]-\mathcal{N}[\hat{u},\bm{x},t]$ over the interval $[0,s]$ to obtain

[TABLE]

Furthermore, we obtain

[TABLE]

Applying the Lipschitz condition (4.8) pointwise for each $s$ yields

[TABLE]

Here, $\delta(\bm{x},t)=u(\bm{x},t)-\hat{u}(\bm{x},t)$ represents the approximation error between the DDPINNs approximated solution and the exact solution. Inequality (4.9) shows that the error $e(\bm{x},s)$ of DDC-PINNs at time $s$, measured in the $Y$-norm, is bounded by the time‑integrated spatial error of the DDPINNs approximation method. For dissipative problems (e.g., the heat equation), $\mathcal{N}$ is linear and globally Lipschitz. For nonlinear problems (e.g., Burgers or KdV), $\mathcal{N}$ is locally Lipschitz on bounded sets. Since the exact solution $u$ is bounded (by well-posedness) and the DDPINNs approximation $\hat{u}$ is accurate, both $u$ and $\hat{u}$ lie in a common bounded set where the Lipschitz condition holds uniformly. Consequently, the error estimate $\left\|e(t)\right\|_{Y}\leq L\int_{0}^{t}\left\|\delta(s)\right\|_{X}ds$ ensures that the DDC-PINNs solution depends continuously on $\hat{u}$ and converges to $u$ as $\|\delta\|\to 0$. Time integration is performed with the explicit RK4 method ($\Delta t=0.001$); because the ODEs right-hand side is independent of $\phi$, the scheme is non‑stiff and stable. No instability was observed in any numerical test.

5 Numerical experiments

In this section, to validate the effectiveness of our proposed method—which combines the flexibility of domain decomposition whilst preserving causal relationships in the solution—we will compare DDC-PINNs with PINNs, XPINNs and IDPINNs through four examples. In these comparisons, certain parameter settings for XPINNs and IDPINNs were selected based on the configuration adopted by DDC-PINNs, whilst the total number of training points and other parameters for PINNs were set to match those of DDC-PINNs. In addition, we compared our method with the neural Galerkin method and CEENs in the same example and verified the superior accuracy of our method in capturing the causality of the solution.

In experiments involving DDC-PINNs and neural Galerkin methods, we use the fourth-order Runge-Kutta method with a step size of 0.001, while for the CEENs method, the time step is set to 0.001. Meanwhile, all neural networks employed the $\tanh(x)$ activation function, with the maximum number of iterations set to 20,000, and the Adam optimizer was used with a learning rate of 0.001. The relative $L^{2}$ error is defined as

[TABLE]

here, $N$, $\hat{u}$ and $u_{\text{ref}}$ denote the total number of computational points, the approximate solution, and the exact solution or reference solution, respectively.

All runtime statistics are computed on the same hardware, a consumer-grade laptop (Lenovo XiaoXin Air 15 ALC 2021) equipped with an AMD Ryzen 7 5700U CPU (8 cores, 16 threads), 16 GB of DDR4 RAM (2666 MHz, onboard), and integrated AMD Radeon Graphics (Vega 8).

Example 5.1**.**

Diffusion problem

[TABLE]

The exact solution of problem (5.2)-(5.4) is $u(x,t)=e^{-t}\sin(\pi x)$.

In the experimental setup, the spatial domain $[-1,1]$ is partitioned into two subdomains $[-1,0]$ and $[0,1]$, leading to two space-time regions$[-1,0]\times[0,1]$ and $[0,1]\times[0,1]$. In the DDC-PINNs method, different neural network architectures are employed for these two regions: the network for the first region consists of 4 hidden layers with 50 neurons each, while the network for the second region consists of 4 hidden layers with 60 neurons each. The network architectures for XPINNs and IDPINNs are the same as that of DDC-PINNs. For the neural Galerkin method and CEENs, a unified neural network architecture with 4 hidden layers and 50 neurons per layer is adopted. The loss function weights are assigned as follows: $\lambda_{1}=1$, $\lambda_{2}=22$, $\lambda_{3}=1$, $\lambda_{4}=1$, $\lambda_{5}=10$,$\lambda_{6}=0.001$, $\lambda_{7}=10$, $\lambda_{8}=0.001$, $\lambda_{9}=1$. Training Point Allocation for XPINNs, IDPINNs, and DDC-PINNs (First Stage):

Each subdomain contains 3000 points.

• 2.

An additional 1000 points are allocated to the subdomain interface.

• 3.

200 points enforce boundary conditions.

• 4.

80 points enforce initial conditions.

Here, for the CEENs, neural Galerkin methods, and DDC-PINNs methods (Stages 2), the spatial grid consists of $200$ points, and the time step is $0.001.$

It is shown in Figure 2 that the exact solution, the DDC-PINNs solution, and the point-wise absolute error for the problem (5.2) -(5.4). In Figure 3, we compare the DDC-PINNs solution and the exact solution in the spatial domain at different time points. In Table 1, we summarize the relative $L^{2}$ errors obtained by using PINNs and the domain decomposition methods (XPINNs, IDPINNs, and DDC-PINNs) for Example 5.1. It can be observed that DDC-PINNs achieve the best performance, yielding the smallest errors. In Table 2, we compare the relevant errors and running times of the neural Galerkin method with time evolution, the CEENs method, and the DDC-PINNs method. In the comparison between DDC-PINNs and the neural Galerkin method, although DDC-PINNs incur a slightly higher computational time cost, they outperform the neural Galerkin method in terms of accuracy. Moreover, compared with the CEENs method, DDC-PINNs are superior in both accuracy and computational time.

Example 5.2**.**

Inviscid Burgers’ equation

[TABLE]

In this example, the reference solution is computed by the finite difference method with central differencing for both temporal ($\Delta t=0.0001$) and spatial ($\Delta x=0.01$) discretizations. The spatial domain $[-1,1]$ is divided into two subdomains, $[-1,0]$ and $[0,1]$. In the DDC-PINNs method, the same neural network architecture is employed for both space-time regions, consisting of 3 hidden layers with 40 neurons each. The weights in the loss function are set as $\lambda_{1}=5$, $\lambda_{2}=20$, $\lambda_{3}=10$, $\lambda_{4}=0.01$, $\lambda_{5}=0.0002$, $\lambda_{6}=0.01$, $\lambda_{7}=0.01$, $\lambda_{8}=0.01$, $\lambda_{9}=0.01$. In addition, both the CEENs method and the neural Galerkin method adopt a network architecture with 4 hidden layers and 50 neurons per layer. Regarding the training point allocation for XPINNs, IDPINNs, and DDC-PINNs (first stage):

Each subdomain is assigned 2000 points.

• 2.

The subdomain interface is given an additional 2000 points.

• 3.

Boundary conditions are enforced by 200 points.

• 4.

Initial conditions are established through 40 points.

Here, for the CEENs, neural Galerkin method, and DDC‑PINNs (stages 2), the number of spatial discretization points is 200 and the time step size is 0.001.

It is shown in Figure 4 that the DDC-PINNs solution, the reference solution, and the point-wise errors between the two. The comparative results at different time points are provided in Figure 5. As shown in Table 3, we report the relative $L^{2}$ errors obtained with PINNs and three domain decomposition methods (XPINNs, IDPINNs, and DDC-PINNs) for Example 5.2. Among these, DDC-PINNs achieve the best performance, giving the smallest errors. Table 4 provides a comparison of the errors and computational times for the time‑dependent neural Galerkin method, the CEENs method, and DDC-PINNs. In direct comparison, DDC-PINNs incur a marginally higher runtime than the neural Galerkin method, yet they deliver higher accuracy. Furthermore, relative to the CEENs method, DDC-PINNs demonstrate advantages in both accuracy and computational efficiency.

Example 5.3**.**

Korteweg-de Vries problem Alexander

[TABLE]

Problems(5.8)-(5.10) have the exact solution $u(x,t)=3sech^{2}(0.5x-0.5t)$.

For DDC-PINNs, the spatial domain $[-40,40]$ is decomposed into$[-40,0]$ and $[0,40]$, with a neural network of two hidden layers (20 neurons each) applied to both subdomains and loss weights $\lambda_{1}=1$, $\lambda_{2}=10$, $\lambda_{3}=1$, $\lambda_{4}=1$, $\lambda_{5}=10$, $\lambda_{6}=0.001$, $\lambda_{7}=0.01$, $\lambda_{8}=0.01$, $\lambda_{9}=0.001$. The CEENs and neural Galerkin methods share the same network architecture. The setup of training point allocation for XPINNs, IDPINNs, and DDC-PINNs (first stage) is as follows:

A total of 3000 training points are assigned to each subdomain.

• 2.

An additional 1000 points are placed along the subdomain interface.

• 3.

Boundary conditions are enforced using 200 points.

• 4.

Initial conditions are imposed via 200 points.

For the CEENs method, the neural Galerkin method, and the second stage of DDC-PINNs, the spatial discretization employs 200 points, and the time step size is set to 0.001.

As shown in Figure 6, the figure presents the DDC-PINNs solution, the exact solution and their point-wise errors . Figure 7 compares the exact and predicted solutions at different time instants. Table 5 compares the relative $L^{2}$ errors obtained by PINNs and three domain decomposition methods (DDC-PINNs, XPINNs and IDPINNs) when solving Example 5.3. DDC-PINNs not only effectively capture the causal relationships within the solution, but also achieve higher accuracy than PINNs, XPINNs and IDPINNs. Furthermore, Table 6 indicates that, although DDC-PINNs require longer computation times than neural Galerkin methods, they demonstrate the best accuracy in capturing the causal relationships in the solution, whilst also outperforming the CEENs method in terms of both computational accuracy and time.

Example 5.4**.**

2D-Heat problem

[TABLE]

$\Omega=[0,\pi]\times[0,\pi]$, The exact solution to problem (5.11)–(5.13) takes the form $u(x,t)=e^{-t}\sin(x)\sin(y)$.

The loss function weights used in DDC-PINNs are $\lambda_{1}=10$, $\lambda_{2}=5$, $\lambda_{3}=5$, $\lambda_{4}=0.03$, $\lambda_{5}=5$, $\lambda_{6}=0.03$, $\lambda_{7}=5$, $\lambda_{8}=5$, and $\lambda_{9}=0.03$. The spatial domain $[0,\pi]\times[0,\pi]$ is divided into two subdomains, $[0,\pi/2]\times[0,\pi]$ and $[\pi/2,\pi]\times[0,\pi]$, with the same network architecture (four hidden layers, $32$ neurons per layer) used in each subdomain for DDC-PINNs. The neural network architectures used in CEENs and the neural Galerkin method also employ a neural network with four hidden layers, each containing $32$ neurons. For the three methods under consideration—XPINNs, IDPINNs, and DDC-PINNs (first stage)—the training point allocation is configured as:

Each subdomain: 5000 points.

• 2.

Subdomain interface: 1000 points.

• 3.

Boundary conditions: 6400 points.

• 4.

Initial conditions: 1600 points.

For the CEENs method, the neural Galerkin method, and the second stage of DDC-PINNs, the spatial discretization uses 2500 points, and the time step is fixed at 0.001.

Figure 8 presents a comparison between the exact solution and the DDC-PINNs solution for problems (5.11)–(5.13) at $t=0.4$ and $t=0.8$, along with the pointwise absolute errors between the exact and predicted solutions at these time instants. Table 7 summarizes the relative $L^{2}$ errors obtained by PINNs and the domain decomposition methods XPINNs, IDPINNs, and DDC-PINNs in Example 5.4. It can be observed that DDC-PINN achieves the smallest errors across all evaluated metrics, demonstrating the best performance. Table 8 compares the causal Neural Galerkin method, the CEENs method, and the DDC-PINNs method in terms of relative $L^{2}$ error and runtime. DDC-PINNs exhibit slightly lower accuracy than the Neural Galerkin method but require significantly less computational cost. Compared with the CEENs method, DDC-PINNs show superior performance in both runtime and accuracy.

6 Conclusions

This paper proposes a novel framework called DDC-PINNs, which combines domain decomposition with classical time integration to improve the solution of time-dependent PDEs. By enhancing the boundary conditions through the addition of extra smoothing terms to both the solution and the PDEs residual, DDC-PINNs is able to effectively recover smooth and accurate solutions at subdomain boundaries. The framework further simplifies the original PDEs into a system of ODEs and employs classical numerical methods for time evolution, thereby explicitly capturing the causal relationships within the solution. The experimental results indicate that DDC-PINNs outperform PINNs, XPINNs, IDPINNs and CEENs in terms of accuracy and computational efficiency; when compared with the neural Galerkin method, they demonstrate varying advantages in terms of computational accuracy and efficiency depending on the specific problem. Currently, this method is applicable to the spatial square domains, and in the future we will consider the more complex spatial domains.

CRediT authorship contribution statement

Xun Yang: Writing-review & editing, Writing-original draft, Methodology, Investigation. Guanqiu Ma: Writing-review & editing, Supervision, Methodology. Maohua Ran: Writing-review & editing, Supervision, Methodology, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Ackonwledgments

This work is supported by the Sichuan Science and Technology Programs (No. 2024NSFSC0441).

Data availability

No data was used for the research described in the article.