Machine Learning Method for Approximate Solutions for Reaction-Diffusion Equations with Multivalued Interaction Functions

This paper presents machine learning methods for approximate solutions of reaction-diffusion equations with multivalued interaction functions. This approach addresses the challenge of finding all possible solutions for such equations, which often lack uniqueness. The proposed method utilizes physics-informed neural networks (PINNs) to approximate generalized solutions.

Keywords:

Subject: Computer Science and Mathematics - Computational Mathematics

1. Introduction

In this paper we establish machine learning methods for approximate solutions of classes of reaction-diffusion equations with multivalued interaction functions allowing for non-unique solutions of the Cauchy problem. The relevance of this problem is primarily due to the lack of methods for finding all solutions for such mathematical objects. Therefore, there is an expectation that another approximate method will provide us with yet another solution for these problems. In addition, methods for approximate solutions of nonlinear systems with partial derivatives without uniqueness are mostly theoretical and are used primarily in qualitative research [1,2]. The availability of computational power for parallel computations and the creation of open-source software libraries such as PyTorch [3] have stimulated a new wave of development in IT and artificial intelligence methods. Sample-based methods for approximate solutions of such problems were first proposed in [4]. To date, such systems with smooth nonlinearities have been qualitatively and numerically studied. There is a need to develop a methodology for approximating generalized solutions of nonlinear differential-operator systems without uniqueness using recurrent neural networks, sample-based methods, and variations of the Monte Carlo method.

Let

T, ν > 0,

and

u_{0} : R^{2} \to R

be a sufficiently smooth function. We consider the problem:

\frac{\partial u}{\partial t} (x, t) \in ν \nabla^{2} u (x, t) - f (u (x, t)), (x, t) \in R^{2} \times [0, T],

(1)

with initial conditions:

u (x, 0) = u_{0} (x), x \in R^{2},

(2)

where

f (s) : = \{\begin{matrix} {0}, & s < 0; \\ [0, 1], & s = 0; \\ {1}, & s > 0 . \end{matrix}

(3)

For a fixed

u_{0} \in C_{0}^{\infty} (R^{2})

let

Ω \subset R^{2}

be a bounded domain with sufficiently smooth boundary and

supp u_{0} \subset Ω .

According to [1] (see the book and references therein), there exists a weak solution

u = u (x, t) \in L^{2} (0, T; H_{0}^{1} (Ω))

with

\frac{\partial u}{\partial t} \in L^{2} (0, T; H^{- 1} (Ω)),

of Problem (1)–(2) in the following sense:

- \int_{0}^{T} \int_{Ω} u (x, t) v (x) η_{t} (t) d x d t + \int_{0}^{T} \int_{Ω} (\nabla u (x, t) \cdot \nabla v (x) + d (x, t) v (x)) η (t) d t = 0,

(4)

for all

v \in C_{0}^{\infty} (Ω), η \in C_{0}^{\infty} (0, T),

where

d : R \times [0, T] \to R

be a measurable function such that

d (x, t) \in f (u (x, t)) f o r a . e . (x, t) \in R^{2} \times (0, T) .

(5)

Such inclusions with multivalued nonlinearities appear in problems of climatology (Budyko-Sellers Model), chemical kinetics (Belousov-Zhabotinsky equations), biology (Lotka–Volterra systems with diffusion), quantum mechanics (FitzHugh–Nagumo system), engineering and medicine (several syntheses and impulse control problems); see [1,2] and references therein.

The main goal of this paper is to develop an algorithm for approximation of solutions for classes of reaction-diffusion equations with multivalued interaction functions allowing for non-unique solutions of the Cauchy problem (1)–(2) via the so-called physics-informed neural networks (PINNs); [5,6,7] and references therein.

2. Methodology of Approximate Solutions for Reaction-Diffusion Equations with Multivalued Interaction Functions

Fix an arbitrary

T > 0,

and a sufficiently smooth function

u_{0} : R \to R .

We approximate the function f by the following Lipschitz functions:

f_{k} (s) : = \{\begin{matrix} 0, & s < 0; \\ k s, & s \in [0, \frac{1}{k}); \\ 1, & s \geq \frac{1}{k}, \end{matrix} k = 1, 2, \dots .

(6)

For a fixed

k = 1, 2, \dots,

consider the problem:

\frac{\partial u_{k}}{\partial t} (x, t) = ν \nabla^{2} u_{k} (x, t) - f_{k} (u_{k} (x, t)), (x, t) \in R^{2} \times [0, T],

(7)

with initial conditions:

u_{k} (x, 0) = u_{0} (x), x \in R^{2} .

(8)

According to [2] and references therein, for each

k = 1, 2, \dots

Problem (7)–(8) has an unique solution

u_{k} \in C^{2, 1} (R^{2} \times [0, T]) .

Moreover, [8] implies that each convergent subsequence

{u_{k_{l}}}_{l = 1, 2, \dots} \subset {u_{k}}_{k = 1, 2, \dots}

of corresponding solutions to Problem (7)–(8) weakly converges to a solution u of Problem (1)–(2) in the space

W : = {z \in L^{2} (0, T; H_{0}^{1} (Ω)) : \frac{\partial z}{\partial t} \in L^{2} (0, T; H^{- 1} (Ω))}

(9)

endowed with the standard graph norm, where

Ω \subset R^{2}

is a bounded domain with sufficiently smooth boundary and

supp u_{0} \subset Ω .

Thus, the first step of the algorithm is to replace the function f in Problem (1)–(2) with

f_{k}

considering Problem (7)–(8) for sufficiently large

k .

Let us now consider Problem (7)–(8) for sufficiently large

k .

Theorem 16.1.1 from [5] allows us to reformulate Problem (7)–(8) as an infinite dimensional stochastic optimization problem over a certain function space. More exactly, let

t \in C ([0, T]; (0, \infty))

ξ \in C (R^{2}; (0, \infty)),

let

(Ω, F, P)

be a probability space, let

T : Ω \to [0, T]

and

X : Ω \to R^{2}

be independent random variables. Assume for all

A \in B ([0, T])

B \in B (R^{2})

that

P (T \in A) = \int_{A} t (t) d t and P (X \in B) = \int_{B} ξ (x) d x .

Note that

f_{k} : R \to R

be Lipschitz continuous, and let

L_{k} : C^{1, 2} (R^{2} \times [0, T], R) \to [0, \infty]

satisfy for all

v = {(v (x, t))}_{(x, t) \in [0, T] \times R^{2}} \in C^{2, 1} (R^{2} \times [0, T])

that

L_{k} (v) = E [{|v (X, 0) - u_{0} (X)|}^{2} + {|\frac{\partial v}{\partial t} (X, T) - ν \nabla^{2} v (X, T) - f_{k} (v (X, T))|}^{2}] .

Theorem 16.1.1 from [5] implies that the following two statements are equivalent:

It holds that $L_{k} (u_{k}) = {inf}_{v \in C^{2, 1} (R^{2} \times [0, T])} L_{k} (v)$ .
It holds $u_{k} \in C^{2, 1} (R^{2} \times [0, T])$ is the solution of Problem (7)–(8).

Thus, the second step of the algorithm is to reduce the regularized Problem (7)–(8) to the infinite dimensional stochastic optimization problem in

C^{2, 1} (R^{2} \times [0, T]) :

\{\begin{matrix} L_{k} (v) \to min, \\ v \in C^{2, 1} (R^{2} \times [0, T]) . \end{matrix}

(10)

However, due to its infinite dimensionality, the optimization problem (10) is not yet suitable for numerical computations. Therefore, we apply the third step, the so-called Deep Galerkin Method (DGM) [9], that is, we transform this infinite dimensional stochastic optimization problem into a finite dimensional one by incorporating artificial neural networks (ANNs); see [5,9] and references therein. Let

a : R \to R

be differentiable, let

h \in N, l_{1}, l_{2}, . . ., l_{h}, d \in N

satisfy

d = 4 l_{1} + [\sum_{k = 2}^{h} l_{k} (l_{k - 1} + 1)] + l_{h} + 1

, and let

L_{k, h} : R^{d} \to [0, \infty)

satisfy for all

θ \in R^{d}

that

\begin{matrix} L_{k, h} (θ) & = L (N_{M_{a, l_{1}}, M_{a, l_{2}}, . . ., M_{a, l_{h}}, {id}_{R}}^{θ, 3}) \\ = E [{|N_{M_{a, l_{1}}, \dots, M_{a, l_{h}}, {id}_{R}}^{θ, 3} (0, X) - g (X)|}^{2} \\ + |\frac{\partial N_{M_{a, l_{1}}, \dots, M_{a, l_{h}}, {id}_{R}}^{θ, 3}}{\partial t} (T, X) - ν \nabla^{2} N_{M_{a, l_{1}}, \dots, M_{a, l_{h}}, {id}_{R}}^{θ, 3} (T, X) \\ {- f_{k} (N_{M_{a, l_{1}}, \dots, M_{a, l_{h}}, {id}_{R}}^{θ, 3} (T, X))|}^{2}], \end{matrix}

(11)

where

M_{ψ, d}

is the d-dimensional version of a function

ψ,

that is,

M_{ψ, d} : R^{d} \to R^{d}

is the function which satisfies for all

x = {(x_{k})}_{k \in {1, 2, \dots, d}} \in R^{d}

y = {(y_{k})}_{k \in {1, 2, \dots, d}} \in R^{d}

with

\forall k \in {1, 2, \dots, d} :

y_{k} = ψ (x_{k})

that

M_{ψ, d} (x) = y;

for each

d, L \in N, l_{0}, l_{1}, . . ., l_{L} \in N, θ \in R^{d}

satisfying

d \geq \sum_{k = 1}^{L} l_{k} (l_{k - 1} + 1),

and for a function

Ψ_{k} : R^{l_{k}} \to R^{l_{k}},

k \in {1, 2, . . ., L},

we denote by

N_{Ψ_{1}, Ψ_{2}, . . ., Ψ_{L}}^{θ, l_{0}} : R^{l_{0}} \to R^{l_{L}}

the realization function of the fully-connected feedforward artificial neural network associated to

θ

with

L + 1

layers with dimensions

(l_{0}, l_{1}, . . ., l_{L})

and activation functions

(Ψ_{1}, Ψ_{2}, . . . Ψ_{L}),

defined as:

\begin{matrix} N_{Ψ_{1}, Ψ_{2}, . . ., Ψ_{L}}^{θ, l_{0}} (x) = (Ψ_{L} \circ A_{l_{L}, l_{L - 1}}^{θ, \sum_{k = 1}^{L - 1} l_{k} (l_{k - 1} + 1)} \circ Ψ_{L - 1} \circ A_{l_{L - 1}, l_{L - 2}}^{θ, \sum_{k = 1}^{L - 2} l_{k} (l_{k - 1} + 1)} \circ . . . \\ . . . \circ Ψ_{2} \circ A_{l_{2}, l_{1}}^{θ, l_{1} (l_{0} + 1)} \circ Ψ_{1} \circ A_{l_{1}, l_{0}}^{θ, 0}) (x), \end{matrix}

for all

x \in R^{l_{0}};

and for each

d, m, n \in N,

s \in N_{0} : = N \cup {0},

θ = (θ_{1}, θ_{2}, \dots, θ_{d}) \in R^{d}

satisfying

d \geq s + m n + m,

the affine function

A_{s, m, n}^{θ}

from

R^{n}

R^{m}

associated to

(θ, s),

is defined as

A_{s, m, n}^{θ} (x) = (\begin{matrix} θ_{s + 1} & θ_{s + 2} & \dots & θ_{s + n} \\ θ_{s + n + 1} & θ_{s + n + 2} & \dots & θ_{s + 2 n} \\ θ_{s + 2 n + 1} & θ_{s + 2 n + 2} & \dots & θ_{s + 3 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ θ_{s + (m - 1) n + 1} & θ_{s + (m - 1) n + 2} & \dots & θ_{s + m n} \end{matrix}) (\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}) + (\begin{matrix} θ_{s + m n + 1} \\ θ_{s + m n + 2} \\ ⋮ \\ θ_{s + m n + m} \end{matrix})

for all

x = (x_{1}, x_{2}, \dots, x_{n}) \in R^{n} .

The final step in the derivation involves approximating the minimizer of

L_{k, h}

using stochastic gradient descent optimization methods [5]. Let

ξ \in R^{d}

J \in N

{(γ_{n})}_{n \in N} \subseteq [0, \infty),

for each

n \in N,

j \in {1, 2, . . ., J}

let

T : Ω \to [0, T]

and

X_{n, j} : Ω \to R^{2}

be random variables. Let for each

n \in N,

j \in {1, 2, . . ., J},

A \in B ([0, T]),

B \in B (R^{2})

P (T \in A) = P (T_{n, j} \in A) and P (X \in B) = P (X_{n, j} \in B) .

(12)

Let

ℓ_{k, h} : R^{d} \times R^{2} \times [0, T] \to R

is defined as

\begin{matrix} ℓ_{k, h} (θ, x, t) = & {|N_{M_{a, l_{1}}, M_{a, l_{2}}, \dots, M_{a, l_{h}}, {id}_{R}}^{θ, 3} (x, 0) - u_{0} (x)|}^{2} \\ + |\frac{\partial N_{M_{a, l_{1}}, M_{a, l_{2}}, \dots, M_{a, l_{h}}, {id}_{R}}^{θ, 3}}{\partial t} (x, t) - ν \nabla^{2} N_{M_{a, l_{1}}, M_{a, l_{2}}, \dots, M_{a, l_{h}}, {id}_{R}}^{θ, 3} (x, t) \\ {- f_{k} (N_{M_{a, l_{1}}, M_{a, l_{2}}, \dots, M_{a, l_{h}}, {id}_{R}}^{θ, 3} (x, t))|}^{2}, \end{matrix}

(13)

for each

θ \in R^{d},

x \in R^{2},

t \in [0, T],

and let

Θ = {(Θ_{n})}_{n \in N_{0}} : N_{0} \times Ω \to R^{d}

satisfy for all

n \in N

that

Θ_{0} = ξ and Θ_{n} = Θ_{n - 1} - γ_{n} [\frac{1}{J} \sum_{j = 1}^{J} (\nabla_{θ} ℓ_{k, h}) (Θ_{n - 1}, T_{n, j}, X_{n, j})] .

(14)

Ultimately, for sufficiently large

k, h, n \in N,

the realization

N_{M_{a, l_{1}}, M_{a, l_{2}}, \dots, M_{a, l_{h}}, {id}_{R}}^{Θ_{n}, 3}

is chosen as an approximation:

N_{M_{a, l_{1}}, M_{a, l_{2}}, \dots, M_{a, l_{h}}, {id}_{R}}^{Θ_{n}, 3} \approx u

of the unknown solution u of (1)–(2) in the space W defined in (9).

So, the following theorem is justified.

Theorem 1.

Let

T > 0,

and

u_{0} \in C_{0}^{\infty} (R^{2}) .

Then the sequence of

{N_{M_{a, l_{1}}, M_{a, l_{2}}, \dots, M_{a, l_{h}}, {id}_{R}}^{Θ_{n}, 3}}_{k, h, n}

defined in (13)–(14) has an accumulation point in the weak topology of W defined in (9). Moreover, each partial limit of the sequence in hands is weakly converges in W to the solution of Problem (1)–(2) in the sense of (4)–(5).

Proof.

According to Steps 1–4 above, to derive PINNs, we approximate u in the space W defined in (9) by a deep ANN

N_{θ} : R^{2} \times [0, T] \to R

with parameters

θ \in R^{d}

and minimize the empirical risk associated to

L_{k} (v)

over the parameter space

R^{d}

. More precisely, we approximate the solution u of (1)–(2) by

N_{θ^{*}}

where

θ^{*} \in arg min_{θ \in R^{d}} \frac{1}{n} \sum_{i = 1}^{n} [{|N_{θ} (X_{i}, 0) - u_{0} (X_{i})|}^{2} + {|\frac{\partial N_{θ}}{\partial t} (X_{i}, T_{i}) - ν \nabla^{2} N_{θ} (X_{i}, T_{i}) - f_{k} (N_{θ} (X_{i}, T_{i}))|}^{2}]

for a suitable choice of training data

{(X_{i}, T_{i})}_{i = 1}^{n}

. Here

n \in N

denotes the number of training samples and the pairs

(X_{i}, T_{i})

i \in {1, 2, \dots, n},

denote the realizations of the random variables X and T.

Analogously, to derive DGMs, we approximate u by a deep Galerkin method (DGM)

G_{θ} : R^{2} \times [0, T] \to R

with parameters

θ \in R^{d}

and minimize the empirical risk associated to

L_{k, h} (v)

over the parameter space

R^{d} .

More precisely, we approximate the solution u of (1)–(2) by

G_{θ^{*}},

where

θ^{*} \in arg min_{θ \in R^{d}} \frac{1}{n} \sum_{i = 1}^{n} [{|G_{θ} (X_{i}, 0) - u_{0} (X_{i})|}^{2} + {|\frac{\partial G_{θ}}{\partial t} (X_{i}, T_{i}) - ν \nabla^{2} G_{θ} (X_{i}, T_{i}) - f_{k} (G_{θ} (X_{i}, T_{i}))|}^{2}]

for a suitable choice of training data

{(X_{i}, T_{i})}_{i = 1}^{n}

. Here

n \in N

denotes the number of training samples and the pairs

(X_{i}, T_{i})

i \in {1, 2, \dots, n},

denote the realizations of the random variables X and T. □

The empirical risk minimization problems for PINNs and DGMs are typically solved using SGD or variants thereof, such as Adam [5]. The gradients of the empirical risk with respect to the parameters

θ

can be computed efficiently using automatic differentiation, which is commonly available in deep learning frameworks such as TensorFlow and PyTorch. We provide implementation details and numerical simulations for PINNs and DGMs in the next section.

3. Numerical Implementation

Let us present a straightforward implementation of the method as detailed in the previous Section for approximating a solution

u \in W

of Problem (1)–(2) with the initial condition

u_{0} (x) : = ψ (x_{1}^{2} + x_{2}^{2}),

where

ψ (s) : = \{\begin{matrix} sin (8 π exp (1 - \frac{3}{3 - s}), & s \in [0, 3); \\ 0, & othervise, \end{matrix}

(15)

(x_{1}, x_{2}) \in R^{2} .

Let

k = 0.01 .

This implementation follows the original proposal by [6], where

20.000

realizations of the random variable

(X, T)

are first chosen. Here, T is uniformly distributed over

[0, 3],

and X follows a normal distribution in

R^{2}

with mean

0 \in R^{2}

and covariance

4 I_{2} \in R^{2 \times 2} .

A fully connected feed-forward ANN with 4 hidden layers, each containing 50 neurons, and employing the Swish activation function is then trained. The training process uses batches of size

256,

sampled from the

20.000

preselected realizations of

(X, T) .

Optimization is carried out using the Adam SGD method. A plot of the resulting approximation of the solution u after

20.000

training steps is shown in Figure 1.

4. Conclusions

In this paper, we presented a novel machine learning methodology for approximating solutions to reaction-diffusion equations with multivalued interaction functions, a class of equations characterized by non-unique solutions. The proposed approach leverages the power of physics-informed neural networks (PINNs) to provide approximate solutions, addressing the need for new methods in this domain.

Our methodology consists of four key steps:

Approximation of the Interaction Function: We replaced the multivalued interaction function with a sequence of Lipschitz continuous functions, ensuring the problem becomes well-posed.
Formulation of the Optimization Problem: The regularized problem was reformulated as an infinite-dimensional stochastic optimization problem.
Application of Deep Galerkin Method (DGM): We transformed the infinite-dimensional problem into a finite-dimensional one by incorporating artificial neural networks (ANNs).
Optimization and Approximation: Using stochastic gradient descent (SGD) optimization methods, we approximated the minimizer of the empirical risk, yielding an approximation of the unknown solution.

The numerical implementation demonstrated the effectiveness of the proposed method. We used a fully connected feed-forward ANN to approximate the solution of a reaction-diffusion equation with specific initial conditions. The results showed that the PINN method could approximate solutions accurately, as evidenced by the visual plots.

The key contributions of this paper are as follows:

Development of a Machine Learning Framework: We established a robust framework using PINNs to tackle reaction-diffusion equations with multivalued interaction functions.
Handling Non-Uniqueness: Our method addresses the challenge of non-unique solutions, providing a practical tool for approximating generalized solutions.
Numerical Validation: We provided a detailed implementation and numerical validation, demonstrating the practical applicability of the proposed approach.

Future work could explore the extension of this methodology to other classes of partial differential equations with multivalued interaction functions, as well as further optimization and refinement of the neural network architectures used in the approximation process. The integration of more advanced machine learning techniques and the exploration of their impact on the accuracy and efficiency of the solutions also present promising avenues for research.

Author Contributions

All the authors contributed equally to this work.

Funding

“This research was funded by EIT Manufacturing asbl, 0123U103025, grant: “EuroSpaceHub - increasing the transfer of space innovations and technologies by bringing together the scientific community, industry and startups in the space industry”. The second and the third authors were partially supported by NRFU project No. 2023.03/0074 “Infinite-dimensional evolutionary equations with multivalued and stochastic dynamics”.

Institutional Review Board Statement

The authors have nothing to declare.

Informed Consent Statement

The authors have nothing to declare.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The author have no relevant financial or non-financial interests to disclose.

References

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Zgurovsky, M.Z.; Kasyanov, P.O. Qualitative and quantitative analysis of nonlinear systems; Springer, 2018.
Paszke, A.; Sam, G.; Chintala.; Soumith.; Chanan, G. PyTorch, 2016. Accessed on June 5, 2024.
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Figure 1. Plots for the functions

{[- 3, 3]}^{2} ∋ x \mapsto U (x, t) \in R,

where

t \in {0, 0.6, 1.2, 1.8, 2.4, 3}

and

U \in C (R^{2} \times [0, 3])

is an approximation of the solution u of Problem (1)–(2) with

u_{0} (x) : = ψ (x_{1}^{2} + x_{2}^{2}),

where

ψ

is defined in (15), computed by means of the PINN method as implemented in Source code 1.

Figure 1. Plots for the functions

{[- 3, 3]}^{2} ∋ x \mapsto U (x, t) \in R,

where

t \in {0, 0.6, 1.2, 1.8, 2.4, 3}

and

U \in C (R^{2} \times [0, 3])

is an approximation of the solution u of Problem (1)–(2) with

u_{0} (x) : = ψ (x_{1}^{2} + x_{2}^{2}),

where

ψ

is defined in (15), computed by means of the PINN method as implemented in Source code 1.

Listing 1. Modified version of sourse code from Section 16.3 of 5.

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Machine Learning Method for Approximate Solutions for Reaction-Diffusion Equations with Multivalued Interaction Functions

Abstract

1. Introduction

2. Methodology of Approximate Solutions for Reaction-Diffusion Equations with Multivalued Interaction Functions

3. Numerical Implementation

4. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

MDPI Initiatives

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