problem solving on rational functions

Rational Functions

Solve applied problems involving rational functions.

In Example 2, we shifted a toolkit function in a way that resulted in the function [latex]f\left(x\right)=\frac{3x+7}{x+2}[/latex]. This is an example of a rational function. A rational function is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.

A General Note: Rational Function

A rational function is a function that can be written as the quotient of two polynomial functions [latex]P\left(x\right) \text{and} Q\left(x\right)[/latex].

Example 3: Solving an Applied Problem Involving a Rational Function

A large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is poured into the tank at a rate of 1 pound per minute. Find the concentration (pounds per gallon) of sugar in the tank after 12 minutes. Is that a greater concentration than at the beginning?

Let t  be the number of minutes since the tap opened. Since the water increases at 10 gallons per minute, and the sugar increases at 1 pound per minute, these are constant rates of change. This tells us the amount of water in the tank is changing linearly, as is the amount of sugar in the tank. We can write an equation independently for each:

The concentration, C , will be the ratio of pounds of sugar to gallons of water

The concentration after 12 minutes is given by evaluating [latex]C\left(t\right)[/latex] at [latex]t=\text{ }12[/latex].

This means the concentration is 17 pounds of sugar to 220 gallons of water.

At the beginning, the concentration is

Since [latex]\frac{17}{220}\approx 0.08>\frac{1}{20}=0.05[/latex], the concentration is greater after 12 minutes than at the beginning.

Analysis of the Solution

To find the horizontal asymptote, divide the leading coefficient in the numerator by the leading coefficient in the denominator:

Notice the horizontal asymptote is [latex]y=\text{ }0.1[/latex]. This means the concentration, C , the ratio of pounds of sugar to gallons of water, will approach 0.1 in the long term.

There are 1,200 freshmen and 1,500 sophomores at a prep rally at noon. After 12 p.m., 20 freshmen arrive at the rally every five minutes while 15 sophomores leave the rally. Find the ratio of freshmen to sophomores at 1 p.m.

• Precalculus. Authored by : Jay Abramson, et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download For Free at : http://cnx.org/contents/[email protected].

Rational Functions, Equations, and Inequalities

Rational Functions are just a ratio of two polynomials (expression with constants and/or variables), and are typically thought of as having at least one variable in the denominator (which can never be 0 ).

Note that we talk about how to graph rationals using their asymptotes in the  Graphing Rational Functions, including Asymptotes  section. Also, since  limits  exist with Rational Functions and their asymptotes, limits are discussed here in the  Limits and Continuity  section . Since factoring is so important in algebra, you may want to revisit it first. Remember that we first learned factoring here in the Solving Quadratics by Factoring and Completing the Square section, and more Advanced Factoring can be found here .

Introducing Rational Expressions

Multiplying, dividing, and simplifying rational functions.

Frequently, rational expressions can be simplified by factoring the numerator, denominator, or both, and crossing out factors. They can be multiplied and divided like regular fractions.

Here are some examples. Note that these look really difficult, but we’re just using a lot of steps of things we already know. That’s the fun of math!  Also, note in the last example, we are dividing rationals , so we flip the second and multiply .

Remember that when you cross out factors, you can cross out from the top and bottom of the same fraction, or top and bottom from different factors that you are multiplying. You can never cross out two things on top, or two things on bottom.

Finding the Common Denominator

When we add or subtract two or more rationals, we need to find the least common denominator (LCD) , just like when we add or subtract regular fractions. If the denominators are the same, we can just add the numerators across, leaving the denominators as they are. We then must be sure we can’t do any further factoring:

$ \require{cancel} \displaystyle \frac{2}{{3x}}+\frac{4}{{3x}}=\frac{{(2+4)}}{{3x}}=\frac{{{{{\cancel{6}}}^{2}}}}{{{{{\cancel{3}}}^{1}}x}}=\frac{2}{x};\,\,\,\,x\ne 0$.

Just like with regular fractions, we want to use the factors in the denominators in every fraction, but not repeat them across denominators. When nothing is common, just multiply the factors.

Adding and Subtracting Rationals

 restricted domains of rational functions.

As we’ve noticed, since rational functions have variables in denominators, we must make sure that the denominators won’t end up as “ 0 ” at any point of solving the problem .

Thus, the domain of $ \displaystyle \frac{{x+1}}{{2x(x-2)(x+3)}}$ is $ \{x:x\ne -3,0,2\}$. This means if we ever get a solution to an equation that contains rational expressions and has variables in the denominator (which they probably will!), we must make sure that none of our answers would make any denominator in that equation “ 0 ” . These “answers” that we can’t use are called extraneous solutions . We’ll see this in the first example below.

Solving Rational Equations

When we solve rational equations, we can multiply both sides of the equations by the least common denominator , or LCD (which is $ \displaystyle \frac{{\text{least common denominator}}}{1}$ in fraction form), and not even worry about working with fractions! The denominators will cancel out and we just solve the equation using the numerators. Remember that with quadratics, we need to get everything to one side with 0 on the other side and either factor or use the Quadratic Formula .

Again, think of multiplying the top by what’s missing in the bottom from the LCD .

Rational Inequalities, including Absolute Values

Solving  rational inequalities  are a little more complicated since we are typically multiplying or dividing by variables, and we don’t know whether these are positive or negative. Remember that we have to  change the direction of the inequality when we multiply or divide by negative numbers . When we solve these rational inequalities, our answers will typically be a range of numbers .

Rational Inequalities from a Graph

Solving rational inequalities algebraically using a sign chart.

The easiest way to solve rational inequalities algebraically is using the sign chart method , which we saw here in the Quadratic Inequalities section . A sign chart or sign pattern is simply a number line that is separated into partitions (intervals or regions), with boundary points (called “ critical values “) that you get by setting the factors of the rational function, both in the numerator and denominator, to 0 and solving for $ x$.

With sign charts, you can pick any point in between the critical values , and see if the whole function is positive or negative . Then you just pick that interval (or intervals) by looking at the inequality. Generally, if the inequality includes the $ =$ sign, you have a closed bracket, and if it doesn’t, you have an open bracket. But any factor that’s in the denominator must have an open bracket for the values that make it 0 , since you can’t have 0 in the denominator.

The first thing you have to do is get everything on the left side (if it isn’t already there) and 0 on the right side , since we can see what intervals make the inequality true. We can only have one term on the left side , so sometimes we have to find a common denominator and combine terms.

Also, it’s a good idea to put open or closed circles  on the critical values to remind ourselves if we have inclusive points (inequalities with equal signs, such as $ \le $ and $ \ge $) or exclusive points (inequalities without equal signs, or factors in the denominators ).

Here are a couple that involves solving radical inequalities with absolute values . (You might want to review  Solving Absolute Value Equations and Inequalities   before continuing on to this topic.)

Applications of Rationals

There are certain types of word problems that typically use rational expressions. These tend to deal with rates , since rates are typically fractions (such as distance over time). We also see problems dealing with plain fractions or percentages in fraction form.

Distance/Rate/Time Problems

With rational rate problems, we must always remember: $ \text{Distance}=\text{Rate }\times \,\,\text{Time}$ . It seems most of the problems deal with comparing times or adding times .

Shalini can run 3 miles per hour faster than her sister Meena can walk. If Shalini ran 12 miles in the same time it took Meena to walk 8 miles, what is the speed of each sister in this case ?

Work Problems:

Work problems typically have to do with different people or things working together and alone, at different rates. Instead of distance, we work with jobs (typically, 1 complete job).

I find that usually the easiest way to work these problems is to remember:

$ \displaystyle \frac{{\operatorname{time} \,\text{together}}}{{\text{time alone}}}\,\,+\,…\,\,+\,\,\frac{{\text{time together}}}{{\text{time alone}}}\,\,=\,\,1$    (or whatever part of the job or jobs is done; if they do half the job, this equals $ \displaystyle \frac{1}{2}$; if they do a job twice, this equals 2 ).

(Use a “$ +$” between the terms if working towards the same goal, such as painting a room, and “$ -$” if working towards opposite goals, such as filling and emptying a pool.) “Proof” : For work problems, $ \text{Rate }\times \,\text{Time = }1\text{ Job}$. Add up individuals’ portions of a job with this formula, using the time working with others (time together):

$ \displaystyle \begin{array}{c}\left( {\text{individual rate }\times \text{ time }} \right)\text{+}…\text{+}\left( {\text{individual rate }\times \text{ time }} \right)=1\\\left( {\displaystyle \frac{1}{{\text{individual time – 1 job}}}\,\,\times \,\,\text{time}} \right)\text{+}…\text{+}\left( {\displaystyle \frac{1}{{\text{individual time – 1 job}}}\,\,\times \,\,\text{time}} \right)=1\\\left( {\displaystyle \frac{{\text{time working with others}}}{{\text{individual time – 1 job}}}} \right)\text{+}…\text{+}\left( {\displaystyle \frac{{\text{time working with others}}}{{\text{individual time – 1 job}}}} \right)=1\end{array}$

Also, as explained after the first example below, often you see this formula as $ \displaystyle \frac{\text{1}}{{\text{time alone}}}\,+…+\,\frac{\text{1}}{{\text{time alone}}}\,=\,\frac{\text{1}}{{\text{time together}}}$.

We are actually adding the Work they complete (together and alone) using formula $ \text{Rate }\times \text{ }\text{Time }=\text{ Work}$, where the Time is 1 hour (or whatever the unit is).

Cost Problems

Rational inequality word problem.

Understand these problems, and practice, practice, practice!

For Practice : Use the Mathway  widget below to try a Rational Function  problem. Click on Submit (the blue arrow to the right of the problem) and click on Solve for x  to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps , or Click Here , you can register at Mathway for a free trial , and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to  Graphing Rational Functions, including Asymptotes     – you’re ready! 

Rational Function Problems

These lessons are compiled to help PreCalculus students learn about rational function problems and applications.

Related Pages Simplifying Rational Expressions Graphing Rational Functions PreCalculus Lessons

The following figure shows how to solve rational equations . Scroll down the page for examples and solutions on how to solve rational function problems and applications.

Rational Function Problems - Work And Tank

The video explains application problems that use rational equations. Part 1 of 2.

• Martin can pour a concrete walkway in 6 hours working alone. Victor has more experience and can pour the same walkway in 4 hours working alone. How long will it take both people to pour the concrete walkway working together?
• An inlet pipe can fill a water tank in 12 hours. An outlet pipe can drain the tank in 20 hours. If both pipes are mistakenly left open, how long will it take to fill the tank?

Rational Function Applications - Work And Rate

The video explains application problems that use rational equations. Part 2 of 2.

• One person can complete a task 8 hours sooner than another person. Working together, both people can perform the task in 3 hours. How many hours does it take each person to complete the task working alone?
• The speed of a passenger train is 12 mph faster than the speed of the freight train. The passenger train travels 330 miles in the same time it takes the freight train to travel 270 miles. Find the speed of each train.

Rational Functions Word Problems - Work, Tank And Pipe

Here are a few examples of work problems that are solved with rational equations.

• Sam can paint a house in 5 hours. Gary can do it in 4 hours. How long will it take the two working together?
• Joy can file 100 claims in 5 hours. Stephen can file 100 claims in 8 hours. If they work together, how long will it take to file 100 claims?
• A water tank is emptied through two drains in 50 minutes. If only the larger drain is used, the tank will empty in 85 minutes. How long would it take to empty if only the smaller drain is used?
• One computer can run a sorting algorithm in 24 minutes. If a second computer is used together with the first, it takes 13 minutes. How long would it take the second computer alone?
• Two pipes are filling a tank. One pipe fills three times as fast as the other. With both pipes working, the tank fills in 84 minutes. How long would each pipe take working alone?

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Rational Equations

Solving rational equations.

A rational equation  is a type of equation where it involves at least one rational expression, a fancy name for a  fraction . The best approach to address this type of equation is to eliminate all the denominators using the idea of LCD (least common denominator). By doing so, the leftover equation to deal with is usually either linear or quadratic.

In this lesson, I want to go over ten (10) worked examples with various levels of difficulty. I believe that most of us learn math by looking at many examples. Here we go!

Examples of How to Solve Rational Equations

Example 1: Solve the rational equation below and make sure you check your answers for extraneous values.

Would it be nice if the denominators are not there? Well, we can’t simply vanish them without any valid algebraic step. The approach is to find the Least Common Denominator (also known Least Common Multiple) and use that to multiply both sides of the rational equation. It results in the removal of the denominators, leaving us with regular equations that we already know how to solve such as linear and quadratic. That is the essence of solving rational equations.

• The LCD is [latex]6x[/latex]. I will multiply both sides of the rational equation by [latex]6x[/latex] to eliminate the denominators. That’s our goal anyway – to make our life much easier.
• You should have something like this after distributing the LCD.
• I decided to keep the variable [latex]x[/latex] on the right side. So remove the [latex]-5x[/latex] on the left by adding both sides by [latex]5x[/latex].
• Simplify. It’s obvious now how to solve this one-step equation. Divide both sides by the coefficient of [latex]5x[/latex].
• Yep! The final answer is [latex]x = 2[/latex] after checking it back into the original rational equation. It yields a true statement.

Always check your “solved answers” back into the original equation to exclude extraneous solutions. This is a critical aspect of the overall approach when dealing with problems like Rational Equations and Radical Equations .

Example 2: Solve the rational equation below and make sure you check your answers for extraneous values.

The first step in solving a rational equation is always to find the “silver bullet” known as LCD. So for this problem, finding the LCD is simple.

Here we go.

Try to express each denominator as unique powers of prime numbers, variables and/or terms.

Multiply together the ones with the highest exponents for each unique prime number, variable and/or terms to get the required LCD.

• The LCD is [latex]9x[/latex]. Distribute it to both sides of the equation to eliminate the denominators.
• To keep the variables on the left side, subtract both sides by [latex]63[/latex].
• The resulting equation is just a one-step equation. Divide both sides by the coefficient of [latex]x[/latex].
• That is it! Check the value [latex]x = – \,39[/latex] back into the main rational equation and it should convince you that it works.

Example 3: Solve the rational equation below and make sure you check your answers for extraneous values.

It looks like the LCD is already given. We have a unique and common term [latex]\left( {x – 3} \right)[/latex] for both of the denominators. The number [latex]9[/latex] has the trivial denominator of [latex]1[/latex] so I will disregard it. Therefore the LCD must be [latex]\left( {x – 3} \right)[/latex].

• The LCD here is [latex]\left( {x – 3} \right)[/latex]. Use it as a multiplier to both sides of the rational equation.
• I hope you get this linear equation after performing some cancellations.

Distribute the constant [latex]9[/latex] into [latex]\left( {x – 3} \right)[/latex].

• Combine the constants on the left side of the equation.
• Move all the numbers to the right side by adding [latex]21[/latex] to both sides.
• Not too bad. Again make it a habit to check the solved “answer” from the original equation.

It should work so yes, [latex]x = 2[/latex] is the final answer.

Example 4: Solve the rational equation below and make sure you check your answers for extraneous values.

I hope that you can tell now what’s the LCD for this problem by inspection. If not, you’ll be fine. Just keep going over a few examples and it will make more sense as you go along.

• The LCD is [latex]4\left( {x + 2} \right)[/latex]. Multiply each side of the equations by it.
• After careful distribution of the LCD into the rational equation, I hope you have this linear equation as well.

Quick note : If ever you’re faced with leftovers in the denominator after multiplication, that means you have an incorrect LCD.

Now, distribute the constants into the parenthesis on both sides.

• Combine the constants on the left side to simplify it.
• At this point, make the decision where to keep the variable.
• Keeping the [latex]x[/latex] to the left means we subtract both sides by [latex]4[/latex].
• Add both sides by [latex]3x[/latex].
• That’s it. Check your answer to verify its validity.

Example 5: Solve the rational equation below and make sure you check your answers for extraneous values.

Focusing on the denominators, the LCD should be [latex]6x[/latex]. Why?

Remember, multiply together “each copy” of the prime numbers or variables with the highest powers.

• The LCD is [latex]6x[/latex]. Distribute to both sides of the given rational equation.
• It should look like after careful cancellation of similar terms.
• Distribute the constant into the parenthesis.
• The variable [latex]x[/latex] can be combined on the left side of the equation.
• Since there’s only one constant on the left, I will keep the variable [latex]x[/latex] to the opposite side.
• So I subtract both sides by [latex]5x[/latex].
• Divide both sides by [latex]-2[/latex] to isolate [latex]x[/latex].
• Yep! We got the final answer.

Example 6: Solve the rational equation below and make sure you check your answers for extraneous values.

Whenever you see a trinomial in the denominator, always factor it out to identify the unique terms. By simple factorization, I found that [latex]{x^2} + 4x – 5 = \left( {x + 5} \right)\left( {x – 1} \right)[/latex]. Not too bad?

Finding the LCD just like in previous problems.

Try to express each denominator as unique powers of prime numbers, variables and/or terms. In this case, we have terms in the form of binomials.

Multiply together the ones with the highest exponents for each unique copy of a prime number, variable and/or terms to get the required LCD.

• Before I distribute the LCD into the rational equations, factor out the denominators completely.

This aids in the cancellations of the commons terms later.

• Multiply each side by the LCD.
• Wow! It’s amazing how quickly the “clutter” of the original problem has been cleaned up.
• Get rid of the parenthesis by the distributive property.

You should end up with a very simple equation to solve.

Example 7: Solve the rational equation below and make sure you check your answers for extraneous values.

Since the denominators are two unique binomials, it makes sense that the LCD is just their product.

• The LCD is [latex]\left( {x + 5} \right)\left( {x – 5} \right)[/latex]. Distribute this into the rational equation.
• It results in a product of two binomials on both sides of the equation.

It makes a lot of sense to perform the FOIL method. Does that ring a bell?

• I expanded both sides of the equation using FOIL. You should have a similar setup up to this point. Now combine like terms (the [latex]x[/latex]) in both sides of the equation.
• What’s wonderful about this is that the squared terms are exactly the same! They should cancel each other out. We could have bumped into a problem if their signs are opposite.
• Subtract both sides by [latex]{x^2}[/latex].
• The problem is reduced to a regular linear equation from a quadratic.
• To isolate the variable [latex]x[/latex] on the left side implies adding both sides by [latex]6x[/latex].
• Move all constant to the right.
• Add both sides by [latex]30[/latex].
• Finally, divide both sides by [latex]5[/latex] and we are done.

Example 8: Solve the rational equation below and make sure you check your answers for extraneous values.

This one looks a bit intimidating. But if we stick to the basics, like finding the LCD correctly, and multiplying it across the equation carefully, we should realize that we can control this “beast” quite easily.

Expressing each denominator as unique powers of terms

Multiply each unique terms with the highest power to obtain the LCD

• Factor out the denominators.
• Multiply both sides by the LCD obtained above.

Be careful now with your cancellations.

• You should end up with something like this when done right.
• Next step, distribute the constants into the parenthesis.

This is getting simpler in each step!

I would combine like terms on both sides also to simplify further.

• This is just a multi-step equation with variables on both sides. Easy!
• To keep [latex]x[/latex] on the left side, subtract both sides by [latex]10x[/latex].
• Move all the pure numbers to the right side.
• Subtract both sides by [latex]15[/latex].
• A simple one-step equation.
• Divide both sides by [latex]5[/latex] to get the final answer. Again, don’t forget to check the value back into the original equation to verify.

Example 9: Solve the rational equation below and make sure you check your answers for extraneous values.

Let’s find the LCD for this problem, and use it to get rid of all the denominators.

Express each denominator as unique powers of terms.

Multiply each unique term with the highest power to determine the LCD.

• Factor out the denominators completely
• Distribute the LCD found above into the given rational equation to eliminate all the denominators.
• We reduced the problem into a very easy linear equation. That’s the “magic” of using LCD.

Multiply the constants into the parenthesis.

• Combine similar terms
• Keep the variable to the left side by subtracting [latex]x[/latex] on both sides.
• Keep constants to the right.
• Add both sides by [latex]8[/latex] to solve for [latex]x[/latex]. Done!

Example 10: Solve the rational equation below and make sure you check your answers for extraneous values.

Start by determining the LCD. Express each denominator as powers of unique terms. Then multiply together the expressions with the highest exponents for each unique term to get the required LCD.

So then we have,

• Factor out the denominators completely.
• Distribute the LCD found above into the rational equation to eliminate all the denominators.
• Critical Step : We are dealing with a quadratic equation here. Therefore keep everything (both variables and constants) on one side forcing the opposite side to equal zero.
• I can make the left side equal to zero by subtracting both sides by [latex]3x[/latex].
• At this point, it is clear that we have a quadratic equation to solve.

Always start with the simplest method before trying anything else. I will utilize the factoring method of the form [latex]x^2+bx+c=0[/latex] since the trinomial is easily factorable by inspection .

• The factors of [latex]{x^2} – 5x + 4 = \left( {x – 1} \right)\left( {x – 4} \right)[/latex]. You can check it by the FOIL method .
• Use the Zero Product Property to solve for [latex]x[/latex].

Set each factor equal to zero, then solve each simple one-step equation.

Again, always check the solved answers back into the original equations to make sure they are valid.

You might also like these tutorials:

• Adding and Subtracting Rational Expressions
• Multiplying Rational Expressions
• Solving Rational Inequalities

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Mathematics > Numerical Analysis

Title: computation of zolotarev rational functions.

Abstract: An algorithm is presented to compute Zolotarev rational functions, that is, rational functions $r_n^*$ of a given degree that are as small as possible on one set $E\subseteq\complex\cup\{\infty\}$ relative to their size on another set $F\subseteq\complex\cup\{\infty\}$ (the third Zolotarev problem). Along the way we also approximate the sign function relative to $E$ and $F$ (the fourth Zolotarev problem).

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Ensemble of physics-informed neural networks for solving plane elasticity problems with examples

• Original Paper
• Published: 29 August 2024

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• Aliki D. Mouratidou   ORCID: orcid.org/0000-0002-8382-1263 1 ,
• Georgios A. Drosopoulos 2 , 3 &
• Georgios E. Stavroulakis 1  

Two-dimensional (plane) elasticity equations in solid mechanics are solved numerically with the use of an ensemble of physics-informed neural networks (PINNs). The system of equations consists of the kinematic definitions, i.e. the strain–displacement relations, the equilibrium equations connecting a stress tensor with external loading forces and the isotropic constitutive relations for stress and strain tensors. Different boundary conditions for the strain tensor and displacements are considered. The proposed computational approach is based on principles of artificial intelligence and uses a developed open-source machine learning platform, scientific software Tensorflow, written in Python and Keras library, an application programming interface, intended for a deep learning. A deep learning is performed through training the physics-informed neural network model in order to fit the plain elasticity equations and given boundary conditions at collocation points. The numerical technique is tested on an example, where the exact solution is given. Two examples with plane stress problems are calculated with the proposed multi-PINN model. The numerical solution is compared with results obtained after using commercial finite element software. The numerical results have shown that an application of a multi-network approach is more beneficial in comparison with using a single PINN with many outputs. The derived results confirmed the efficiency of the introduced methodology. The proposed technique can be extended and applied to the structures with nonlinear material properties.

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Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., Kudlur, M., Levenberg, J., Monga, R., Moore, S., Murray, D.G., Steiner, B., Tucker, P., Vasudevan, V., Warden, P., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: a system for large-scale machine learning. In: 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), USENIX Association, Savannah, GA, pp. 265-283. URL https://www.usenix.org/conference/osdi16/technical-sessions/presentation/abadi. (2016)

Baydin, A., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res. 18 , 1 (2018)

MathSciNet   Google Scholar  

Bazmara, M., Mianroodi, M., Silani, M.: Application of physics-informed neural networks for nonlinear buckling analysis of beams. Acta. Mech. Sin. 39 , 422438 (2023)

Article   MathSciNet   Google Scholar  

Cai, S., Mao, Z., Wang, Z., Yin, M., Karniadakis, G.E.: Physics-informed neural networks (PINNs) for fluid mechanics: a review. Acta Mech. Sin. 37 , 1–12 (2022)

Chadha, C., He, J., Abueidda, D., Koric, S., Guleryuz, E., Jasiuk, I.: Improving the accuracy of the deep energy method. Acta Mech. 234 , 5975–5998 (2023)

Chen, Y., Lu, L., Karniadakis, G.E., Dal Negro, L.: Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Opt. Express 28 (8), 11618–11633 (2020)

Article   Google Scholar  

Chollet, F.: Deep learning with python, Manning Publications Company, URL https://books.google.ca/books?id= Yo3CAQAACAAJ (2017)

Drosopoulos, G.A., Stavroulakis, G.E.: Non-linear mechanics for composite. In: Heterogeneous Structures, CRC Press, Taylor and Francis (2022)

Drosopoulos, G.A., Stavroulakis, G.E.: Data-driven computational homogenization using neural networks: FE2-NN application on damaged masonry. ACM J. Comput. Cult. Herit. 14 , 1–19 (2020)

Google Scholar  

Faroughi, S., Darvishi, A., Rezaei, S.: On the order of derivation in the training of physics-informed neural networks: case studies for non-uniform beam structures. Acta Mech. 234 , 5673–5695 (2023)

Faroughi, S.A., Pawar, N., Fernandes, C., Raissi, M., Das, S., Kalantari, N.K., Mahjour, S.K.: Physics-guided, physics-informed, and physics-encoded neural networks. Sci. Comput. (2023). https://doi.org/10.48550/arXiv.2211.07377

Fletcher, R.: Practical methods of optimization, 2nd edn. John Wiley & Sons, New York (1987)

Güne, A., Baydin, G., Pearlmutter, B.A., Siskind, J.M.: Automatic differentiation in machine learning: a survey, J. Mach. Learn. Res. 18 , 1-43, (2018) URL http://www.jmlr.org/papers/volume18/17-468/17-468.pdf

Guo, M., Haghighat, E.: An energy-based error bound of physics-informed neural network solutions in elasticity. J. Eng. Mech. (2020). https://doi.org/10.48550/arXiv.2010.09088

Haghighat, E., Juanes, R.: Sciann: a keras/tensorflow wrapper for scientific computations and physics-informed deep learning using artificial neural networks. Comp. Meth. Appl. Mech. Eng. 373 , 113552 (2021)

Haghighat, E., Raissi, M., Moure, A., Gomez, H., and Juanes, R.: A deep learning framework for solution and discovery in solid mechanics. https://arxiv.org/abs/2003.02751 (2020)

Harandi, A., Moeineddin, A., Kaliske, M., Reese, S., Rezaei, S.: Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains. Int J Numer. Methods Eng. 125 , e7388 (2024)

Kadeethum, T., Jorgensen, T., Nick, H.: Physics-informed neural networks for solving nonlinear diffusivity and Biot’s equations. PLoS ONE 15 , e0232683 (2020)

Karniadakis, G.E., Kevrekidis, I.G., Lu, L., et al.: Physics-informed machine learning. Nat. Rev. Phys. 3 , 422–440 (2021)

Katsikis, D., Muradova, A.D., Stavroulakis, G.S.: A gentle introduction to physics-informed neural networks, with applications in static rod and beam problems. Jr. Adv. Appl. & Comput. Math. 9 , 103–128 (2022)

Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization, computer science, mathematics. In: The International Conference on Learning Representations (ICLR) (2015)

Kortesis, S., Panagiotopoulos, P.D.: Neural networks for computing in structural analysis: methods and prospects of applications. Int. Jr. Numeric. Meth. Eng. 36 , 2305–2318 (1993)

Kovachki, N., Lanthaler, S., Mishra, S.: On universal approximation and error bounds for Fourier neural operator. J. Mach. Learn. Res. 22 , 1–76 (2021)

Lagaris, E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9 , 987–1000 (1998)

Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., Anandkumar, A.: Fourier neural operator for parametric partial differential equations, arXiv preprint arXiv:2010.08895 (2020)

Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Program. 45 , 503–528 (1989)

Lu, L., Jin, P., Pang, G., Zhang, Z., Karniadakis, G.E.: Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat. Mach. Intell. 3 , 218–229 (2021). https://doi.org/10.1038/s42256-021-00302-5

Meade, A.J., Fernandez, A.A.: The numerical solution of linear ordinary differential equations by feed-forward neural networks. Math. Comput. Model. 19 , 1–25 (1994). https://doi.org/10.1016/0895-7177(94)90095-7

Muradova, A.D., Stavroulakis, G.E.: Physics-informed neural networks for elastic plate problems with bending and Winkler-type contact effects. J. Serb. Soc. Comput. Mech. 15 , 45–54 (2021)

Muradova, A.D, Stavroulakis, G.E.: Physics-informed Neural Networks for the solution of unilateral contact problems. In: Book of Proceedings, 13th International Congress on Mechanics HSTAM, pp. 451-459. (2022). https://hstam2022.eap.gr/book-of-proceedings/

Muradova, A.D., Stavroulakis, G.E.: The projective-iterative method and neural network estimation for buckling of elastic plates in nonlinear theory. Commun. Nonlin. Sci. Num. Sim. 12 , 1068–1088 (2007)

Niu, S., Zhang, E., Bazilevs, Y., Srivastava, V.: Modeling finite-strain plasticity using physics-informed neural network and assessment of the network performance. J. Mech. Phys. Solids 172 , 105177 (2023)

Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378 , 686–707 (2019)

Rezaei, S., Harandi, A., Moeineddin, A., Xu, B.-X., Reese, S.: A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method. Comput. Meth. Appl. Mech. Eng. 401 (Part B), 115616 (2022)

Ruder, S.: An overview of gradient descent optimization algorithms (2017) https://arxiv.org/abs/1609.04747

Shin, Y., Darbon, J., Karniadakis, G.E.: On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Commun. Comput. Phys. 28 , 2042–2074 (2020)

Stavroulakis, G.E.: Inverse and crack identification problems in engineering mechanics. Springer, New York (2000)

Stavroulakis, G.E., Avdelas, A., Abdalla, K.M., Panagiotopoulos, P.D.: A neural network approach to the modelling, calculation and identification of semi-rigid connections in steel structures. J. Construct. Steel Res. 44 , 91–105 (1997)

Stavroulakis, G., Bolzon, G., Waszczyszyn, Z., Ziemianski, L.: Inverse analysis. In: Karihaloo, B., Ritchie, R.O., Milne, I. (eds) Comprehensive structural integrity, numerical and computational methods, vol. 3, Chap 13, pp. 685–718. Elsevier, Amsterdam (2003)

Tartakovsky, A. M., Marrero, C. O., Perdikaris, P., Tartakovsky, G. D., and Barajas- Solano D.: Learning parameters and constitutive relationships with physics informed deep neural networks. arXiv preprint arXiv:1808.03398 (2018)

Theocaris, P.S., Panagiotopoulos, P.D.: Plasticity including the Bauschinger effect, studied by a neural network approach. Acta Mech. 113 , 63–75 (1995). https://doi.org/10.1007/BF01212634

Waszczyszyn, Z., Ziemiański, L.: Neural networks in the identification analysis of structural mechanics problems. In: Mróz, Z., Stavroulakis, G.E. (eds.) Parameter identification of materials and structures, CISM International Centre for Mechanical Sciences (Courses and Lectures), p. 469. Springer, Vienna (2005)

Xiao, Y., Wei, Z., Wang, Z.: A limited memory BFGS-type method for large-scale unconstrained optimization. Comput. Math. Appl. 56 , 1001–1009 (2008)

Yagawa, G., Oishi, A.: Computational mechanics with neural networks. Springer, New York (2021)

Book   Google Scholar  

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The work of A.D.M. and G.E.S. has been supported by the Project Safe-Aorta, which was implemented in the framework of the Action “Flagship actions in interdisciplinary scientific fields with a special focus on the productive fabric”, through the National Recovery and Resilience Fund Greece 2.0 and funded by the European Union-NextGenerationEU (Project ID:TAEDR-0535983)

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School of Production Engineering and Management, Institute of Computational Mechanics and Optimization, Technical University of Crete, Kounoupidiana, 73100, Chania, Crete, Greece

Aliki D. Mouratidou & Georgios E. Stavroulakis

Discipline of Civil Engineering, School of Engineering and Computing, University of Central Lancashire, Preston campus, Preston, PR1 2HE, UK

Georgios A. Drosopoulos

Discipline of Civil Engineering, School of Engineering, University of KwaZulu-Natal, Durban campus, Durban, 4041, South Africa

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Mouratidou, A.D., Drosopoulos, G.A. & Stavroulakis, G.E. Ensemble of physics-informed neural networks for solving plane elasticity problems with examples. Acta Mech (2024). https://doi.org/10.1007/s00707-024-04053-3

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Received : 26 February 2024

Revised : 27 May 2024

Accepted : 31 July 2024

Published : 29 August 2024

DOI : https://doi.org/10.1007/s00707-024-04053-3

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