Discovery of subdiffusion problem with noisy data via deep learning

Deep learning has been extended to many different practical fields, including image analysis, natural language processing, system recognizing etc IYA:2016 . There are already some important progress for numerically solving the ODEs and PDEs systems in deep learning. For example, variational methods WEBY:2017 ; YGMK:2021 ; YLJ:2021 ; CGD:2021 , Galerkin methods DGM:2018 ; CJR:2021 , random particle method XUY:2020 , different operators method LZY:2021 , linear multistep method DUQ:2021 . Based on the the neural networks, the data-driven discovery of differential equations has been proposed in RTP:2019 ; SHR:2019 ; DEM:2020 ; MRI:2018 ; QWX:2019 ; DUQ:2021 including the space-fractional differential equations GRPK:19 . In particular, the discovery of traditional ODEs dynamics using linear multistep methods in deep learning have been discussed in DUQ:2021 ; RD:21 and RTP:2019 . Identifying source function from observed data is a meaningful and challenge topic for the time-dependent PDEs.

Over the few decades, fractional models have been attracted wide interest since it can effectively characterize the ubiquitous power-law phenomena with noise data EK:11 ; LWD:17 , which are applied in underground environment problems, transport in turbulent plasma, bacterial motion transport, etc. M2000 . In this work, we study the discovery of the following subdiffusion in deep learning with noise data, whose prototype is LWD:17 ; MS:12 ; WCDBD:22 , for $0<\alpha\leq 1,$

with $A$ a positive definite, selfadjoint, linear Laplacian operator. The Caputo fractional derivative Podlubny:99 is defined by

and $f$ is a given forcing term $\overline{f}$ with noise data. In practice, $f$ is not completely known and is disturbed around some known quantity $\overline{f}$, i.e.,

Here $\overline{f}$ is deterministic and is usually known while no exact behavior exists of the perturbation (noise) term. The uncertainty (lack of information) about $f$ (the perturbation term) is naturally represented as a stochastic quantity, see ZK:17 . In the following, we focus on the uniform noise and Gaussian noise. More general noise ZK:17 such as white noise, Wiener process or Brownian motion (which corresponds to the stochastic fractional PDEs LWD:17 ; WCDBD:22 ) can be similarly studied, since it can be discretized as like uniform noise form by a simple arithmetic operations LWD:17 ; MS:12 ; WCDBD:22 .

The main contribution of this paper is to discover subdiffusion via $L_{1}$ approximation in deep learning with noisy data, where we design two types neural networks to deal with the fractional order $\alpha$, since $\alpha$ is a variable parameter. More concretely, we use two strategies (fixed $\alpha$ and variable $\alpha$, respectively) to train network, which improves the generalization for the subdiffusion problem. The advantage of first Type is that model (4.2) can be trained for each fixed $\alpha$, which reduces the computational count and required storage, since it fixes $\alpha$ as input data. Obviously, it may loss some accuracy. To recover the accuracy, second Type is complemented, which needs more computational count. It implies an interesting generalized structure by combining Type 1 and Type 2, which may keep suitable accuracy and reduces computational count.

The paper is organized as follows. In the next section, we introduce the discretization schemes for the subdiffusion problem (1.1) and its discovery of subdiffusion in Section 3. We construct the discovery of subdiffusion based on the basic deep neural network (DNN) in Section 4. The two types DNN are designed/developed for the subdiffusion model in Section 5. To show the effectiveness of the presented schemes, results of numerical experiments with noisy data are reported in Section 6. Finally, we conclude the paper with some remarks on the presented results.

In this section, we introduce the $L_{1}$ approximation for solving the subdiffusion. Let $t_{n}=nh_{t}$, $n=0,1,...N_{t}$ be a partition of the time interval $[0,T]$ with the grid size $\tau=T/N_{t}$. Denote the mesh points $x_{m}=mh_{x}$, $m=0,1,\cdots,N_{x}$ with the uniform grid size $h_{x}=l/M$, $\Omega=(0,l)$.

The objective for solving the initial value problem in (1.1) is to find the approximation $u_{m}^{n}\approx u(x_{m},t_{n})$ for given source function $f(x_{m},t_{n})$.

The diffusion term is approximated by a standard second-order discretization:

There are serval ways to discretize the Caputo fractional substantial derivative (1.2). For convenience, we use the following standard $L_{1}$ scheme ChJB:21 ; SOG:17 ; Lin:07 with the truncation error $\mathcal{O}\left(\tau^{2-\alpha}\right)$, namely,

Here the coefficients are computed by $\omega^{(\alpha)}_{0}=\frac{1}{\Gamma(2-\alpha)\tau^{\alpha}}$, $\omega^{(\alpha)}_{n}=\frac{(n-1)^{1-\alpha}-n^{1-\alpha}}{\Gamma(2-\alpha)\tau^{\alpha}}$ and

Thus, we approximate (1.1) by the discrete problem with given $f(x_{m},t_{n})=f_{m}^{n}$

The discovery of subdiffusion is essentially an inverse process of solving a subdiffusion problem (1.2). That means, suppose that only the information of $u(x_{m},t_{n})$ at the pairs of the uniform grid points $\left\{x_{m},t_{n}\right\}$ is provided, we need to recover the source function $f$ RD:21 . Assume $u(x,t)$ and $f(x,t)$ both are unknown in subfiffuion system (1.1) with given $u_{m}^{n}=u(x_{m},t_{n})$, the target is to approximate the close-form expression for $f(x_{m},t_{n})$. We can use the approximated source function $f_{m}^{n}$ to rebuild the discrete relation between $f_{m}^{n}$ and $u_{m}^{n}$, namely,

It should be noted that (3.1) directly follows the $L_{1}$ approximation in (2.3). Different from (2.3) that evaluates $u_{m}^{n}$ given $f_{m}^{n}=f(x_{m},t_{n})$, model (3.1) computes $f_{m}^{n}$ from the given data $u_{m}^{n}$. It also indicates the discovery of subdiffusion is actually an inverse process of solving the subdiffusion problem.

In this section, we introduce the basic deep neural network (DNN) CWHZ:21 ; ZuowShen:2021 ; CGD:2021 ; SHR:2019 , which will be used in discovery of subdiffusion.

It is an interesting question how we can train the discovery of subdiffusion with fractional order $\alpha$, since $\alpha$ is variable parameter. Developed the structure of deep neural network in Subsection 4.1, we use two types (fixed $\alpha$ and variable $\alpha$, respectively) of DNN for training the discovery of subdiffusion (4.2) in this work.

In this section, we discover subdiffusion problem (1.1) with uniform noise and Gaussian noise in deep learning. More general noise ZK:17 such as white noise, Wiener process or Brownian motion (which corresponds to the stochastic fractional PDEs LWD:17 ; WCDBD:22 ) can be similarly studied, since it can be discretized as like uniform noise form by a simple arithmetic operations LWD:17 ; MS:12 ; WCDBD:22 . Several examples are provided to show the performance of subdiffusion discovery via $L_{1}$ approximation in DNN by Algorithm 1.

The relative error $||\cdot||_{r}$ is measured by

where we actually use the well-known Frobenius norm of the $n\times n$ matrix $A$, i.e.,

In this work, we firstly propose the discovery of subdiffusion with noisy data in deep learning based on the designing two types. The numerical experiments are given to illustrate the availability using deep learning even for the $50\%$ noise level. The advantage of first Type is that model (4.2) can be trained for each fixed $\alpha$, which reduces the computational count and required storage, since it fixes $\alpha$ as input data. Obviously, it may loss some accuracy. To recover the accuracy, second Type is complemented, which needs more computational count. It implies an interesting generalized structure by combining Type 1 and Type 2, which may keep suitable accuracy and reduces computational count. As a result, even when discovering the interger-order equations, the proposed type DNNs have some advantages compared with the traditional numerical scheme and previous works to solve the data-driven discovery for differential equations. The interesting topic is reducing the computational count and storage by fast multigrid or conjugate gradient squared method CCNWL:20 ; CES:20 . Another interesting topic is how to design the correction of $L_{1}$ approximation for reducing the boundary noise pollution, since subdiffuison model appears a layer or blows up at $t=0$ ShCh:2020 ; SOG:17 with low time regularity.