DiffEq vs. Biology: Which to Choose?

[0rthodontist]

Is introductory differential equations boring? I can take another course and DiffEq is what I am considering mainly because it is mentioned a lot. I have no real need to take this course, unless it is useful in AI. If I take the course it will be taught at a general level in a class of 60, probably mainly to freshmen.
--Is there anything about it that a computer can't do better?
--Does it introduce interesting concepts? I am not talking about advanced courses in DiffEq since I'm not likely to take those.

My alternative is to do a general education requirement such as biology.

 

[franznietzsche]

Diffeq to freshmen? More likely sophmores with a year of calc. My differential equation class was absolutely awful, but the professor was awful and we used what has to be the single worst textbook I have ever seen, in any subject.

 

[neurocomp2003]

DE used in AI depending on the AI you look into, speech&language, ALife, nnets. DE used to study flow. ODE and dynamical systems are better but you do need to learn to solve DEs(which is what a first course in DE teaches...or should).

 

[chroot]

Honestly, a basic AI class is not going to involve solving many differential equations by hand, which is what a Diff Eq class is going to teach you to do. If you're only interested in it for its applications in AI, don't bother.

On the other hand, Diff Eq is used everywhere in computer modeling and computerized systems, and you will be better prepared for a computer-related career if you take it. AI is pretty much worthless career-wise, but it's definitely interesting.

- Warren

 

[mathwonk]

diff eq has traditionally been bprong. martin braun at brown university wrote a book designed to counteract that, and i think he succeeded.

ill try to find you a copy of it.

 

[mathwonk]

try and beat this price for an outstanding book.

Differential equations and their applications: An introduction to applied mathematics (Applied mathematical sciences) (ISBN: 038790266X)
Martin Braun Bookseller: Better World Books
(Mishawaka, IN, U.S.A.) Price: US$ 1.04
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US$ 2.99
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Book Description: Springer-Verlag. Book Condition: Used - Good. Shows some signs of wear, and may have some markings on the inside. 100% Money Back Guarantee.

anyone who pays $130. for a copy of boyce diprima, given this situation, has been warned.

 

[0rthodontist]

Ah, you're right franz, it is aimed at sophomores. It said "5 spots reserved for incoming freshmen" in the course listing which misled me.

I do indeed hope to make AI my career. AI is already in somewhat widespread current application and as the science of making it increases, it can only become more important. It's my opinion that improved AI will be the next major revolution in technology. My question really is though, what major ideas will an intro DE course contain (other than procedures for solving DEs)? While I'm asking, what major ideas do advanced DE's contain?

When I look at some graduate course descriptions for machine learning and neural networks at various schools, I see linear algebra and statistics, not DE's as a requirement. However, I see that nonlinear dynamics, which also interests me whether or not I ever study it, does require DEs. What else requires DE's?

Actually, through a recent turn of events I'm probably not going to take DE's this semester but I will have the option to take a probably better DE course next semester--if it's worth it.

Edit: Thanks Mathwonk, I'll check that out

 

[chroot]

You're a smart guy, Ortho, but be careful about trying to make a career out of AI. If you intend to be a researcher, you'll have a lot of stiff competition for resources. If you intend to work in industry, you will honestly have a hard time finding work.

- Warren

 

[Pythagorean]

My career choice is physics, but I'm absolutely interested in AI. In my sparetime, I try to study the human brain and I have a couple programming courses under my belt (just the basics of java and matlab, nothing impressive), but I give you my best regards in revolutionizing AI. If you don't, then you're going to be right about where chroot put you.

As for the original question on diff eq, I always found it intriguing, because for me, that's what physics is about, how something changes with respect to something else, and being able to manipulate and evolve these concepts mathematically seems like a valuable tool.

My teacher didn't speak good english, and when ever we asked him questions he always just said "it is seemple!". It seemed like he thought USAmericans were just stupid and couildn't be taught much more than to memorize.

Despite that, I still like the subject.

 

[Rach3]

anyone who pays $130. for a copy of boyce diprima, given this situation, has been warned.

I was forced to pay up, by my well-intentioned but thoughtless math professor. Of course this was five years ago, it was a lot less than $130, maybe $80-90 (textbooks have gone way up this decade). I hate doing this - all the more so when in certain cases a $150 text is useless garbage padded with multimedia exercises and full-page photographs.

It's ridiculous to see Boyce&DiPrima selling for three times as much as some Springer hardcovers (and I thought those were expensive).

 

[J77]

diff eq has traditionally been bprong.

but you have to do it to go on to the more interesting nonlinear and pde stuff.

B&DP costs $130 dollars these days!

Still a good book for a first course though.

If you want to keep your options open - like has been mentioned - DEs are found in tonnes of applications in almost every field of science.

AI?

Well - how about obtaining and sorting data from the real world, and using this data, within a DE model, in order to predict future events. For example, traffic, weather... sounds like AI to me - you do know it's not about robots, right?

 

[d_leet]

I think that a class being boring is pretty dependent on the professor teaching the course, that aside I taught myself most of what consists a first course in ordinary differential equations from the new edition of Boyce and Diprema, and yes there were parts that were boring, but I think that is wholly my own fault for not having quite the ambition I should have, but now I'm taking a partial differential equations course at a local college and it is definitely one of, if not the best course I have taken thus far, but I'm pretty sure that this is mainly because the professor teaching this course is one of the best I've had.

 

[moose]

My diff eq class at UA over the summer used a horrible textbook. The prof was OK, but not great by any means. I survived by checking out a few diff eq books from the science/engineering library... It was boring to me, probably because of the prof.

 

[0rthodontist]

What I really want to ask is what does it teach that's good--what is interesting about it. If it's all about manipulating equations with no great ideas then that's what computer algebra systems are for. What are the big ideas?

 

[J77]

What I really want to ask is what does it teach that's good--what is interesting about it. If it's all about manipulating equations with no great ideas then that's what computer algebra systems are for. What are the big ideas?

It's the basis of nonlinear differential equations of many forms - ordinary differential equations, partial differential equations, differential algebraic equations, delay differential equations, integro-differential equations, functional equations.

Which leads onto: bifurcation theory, catastrophe theory.

And more specilist stuff like KAM theory.

Just as examples.

 

[0rthodontist]

Okay--I can imagine that being aware of the existence of those types of equations might qualify as an idea, even if the solution is merely mechanical. Bifurcation theory and catastrophe theory sound interesting but I will probably never study them.

 

[kdinser]

An intro course in diff EQ largely covers how to derive formulas to predict the behavior of a system when you already know some information about that system, mainly it's rate of change and how things can influence that rate of change. Also how to derive formulas when you are given charts of data. You learn other things to like lapace transforms, because they are extremely useful for solving diff EQ's.

The best simple example I can come up with is how to derive the basic kinematics formulas taught in a first year physics course. Of course, 2 or 3 weeks into the semester and you will be well past systems that are this easy to solve.

To me, it was both the most interesting math class I ever took and the most boring. Interesting because I knew how important it was going to be in the rest of my engineering classes. At the same time though, it was boring as hell because any application of the material was largely ignored in favor of mathematical proofs and consideration of extreme details that only a mathematician would be concerned with or find interesting .

 

[Pythagorean]

An intro course in diff EQ largely covers how to derive formulas to predict the behavior of a system when you already know some information about that system, mainly it's rate of change and how things can influence that rate of change. Also how to derive formulas when you are given charts of data. You learn other things to like lapace transforms, because they are extremely useful for solving diff EQ's.

The best simple example I can come up with is how to derive the basic kinematics formulas taught in a first year physics course. Of course, 2 or 3 weeks into the semester and you will be well past systems that are this easy to solve.

To me, it was both the most interesting math class I ever took and the most boring. Interesting because I knew how important it was going to be in the rest of my engineering classes. At the same time though, it was boring as hell because any application of the material was largely ignored in favor of mathematical proofs and consideration of extreme details that only a mathematician would be concerned with or find interesting .

In my mechanics class (third year physics course/unit) my professor said we should have diff eq, than said we'd be learning it over in the physics way. I like being able to see the physical implications of things, personally. The proofs seem abstract and meaningless compared to the physical intepretations.

 

[neurocomp2003]

DEs lead to the study of Ordinary DEs, Partial DEs, Dynamical Systems, Bifurcation Theory & Chaos as well as Classical Mechanics and Fluid Mechanics(all pertaining to the understanding of FLOW/MOTION)

You will have to list some of the AI topics that you are interested in.

As far as NNets go it depends on how naively an Algorithm you are designing. Too my knowledge the simple NNets and their error analysis require some knowledge of DEs not necessarily advanced info on how to solve the equation but to simply understand hwo to minimize error(you could probably learn this in first year calc).

But for knowledge into Pattern Recognition and Speech Recognition & language it would be best ot know PDEs(Transforms).

In ALife: you might want to study the migratory flow of huge systems of agents not just simple 1-10 agents. One way to tackle the method is through studying FLOW itself, which is a DE problem

If you want to design Virtual Worlds to test your Agents in which includes physics-based principles. Understanding Fluid Mechanics would be a good idea.

Logic-based systems don't require DEs...but I believe Adaptive Systems do. Hence both Engineering & Science students normally take DEs(unless in the life sciences).

 

[0rthodontist]

I have had the full basic calculus series (last one called Calc IV here). Actually, Calc IV was the worst course I have ever taken--the most insipid, boring, repetitive thing, taught by someone (a researcher in DiffEq's) who thought little of her class and assigned large amounts of the most routine material the book had to offer, going barely if at all beyond Calc III, which was a great course. I'm mainly looking to avoid taking a course that bad again. If Diff Eq's is a course based only on manual calculation with little intelligence--no.

I am most interested in neural networks, but I already understand their basic operation. I'm not only looking for math that is instantly applicable to a particular field in AI. I'm looking for math that might potentially be applicable, that contains nice ideas that could be relevant. For example, when I took my theory of computation course, there was the state-minimization algorithm. This was a wonderful and interesting thing that is related to both backpropagation and modus tollens--even if I never use this, I am enriched by being aware of it.

 

[neurocomp2003]

There is a whole field of NNets called Spiking Neurons which is more I guess MathNeurosci then Artificial. It takes a more rigourous approach to studying the concept of a Neuron/Networks and requires DEs. Especially PDEs in spatially distributed Nets.

IMO, DEs is a rather easy course to learn on your own(I didnt go to a single class, granted it was at 8 or 9am). If your looking for more advance work...look to a PDE or ODE class(though they might be mind-numbingly easy but may request DE as a prereq).

My best suggestion would be to take some statistical course. A senior level [1] Stochastics(one that teaches Markov Chains), [2]Operations Research(queuing and time slicing) or [3]
Mathematical Statistics (something I regret not taking, its use in rigourous error analysis in NNets, that touches on Bayesian type nets).

I wouldn't take Biology cause its utterly useless(nothing you can't read from a first year text,without failing the course). Take a senior level psych course in vision,language,motor,audition.

May I ask what you plan to do with NNets? You must have a reason for studying them other then you see them as the future of AI.

Also what did CALC IV teach? Vector Calculus? stoke/divergence etc.

If you can an imaging course might be useful as an app for NNets.

Lastly any course that will teach you Markov Chains(if you don't alreayd know it) would be a good course to take, if the prof isn't horrible or doesn't speak english properly.

Markov Chains are used in NNets and Reinforcement Learnig, a field related,if not contained in NNets.

 

[0rthodontist]

There is a whole field of NNets called Spiking Neurons which is more I guess MathNeurosci then Artificial. It takes a more rigourous approach to studying the concept of a Neuron/Networks and requires DEs. Especially PDEs in spatially distributed Nets.

That's the kind of thing I was hoping to see!

IMO, DEs is a rather easy course to learn on your own(I didnt go to a single class, granted it was at 8 or 9am). If your looking for more advance work...look to a PDE or ODE class(though they might be mind-numbingly easy but may request DE as a prereq).

My best suggestion would be to take some statistical course. A senior level [1] Stochastics(one that teaches Markov Chains), [2]Operations Research(queuing and time slicing) or [3]
Mathematical Statistics (something I regret not taking, its use in rigourous error analysis in NNets, that touches on Bayesian type nets).

I've had two semesters of mathematical statistics. I didn't learn a lot from them because my prof was very sick and I didn't really have the prereq for it at the time. My AI class this semester will be doing some Bayesian networks since it is the professor's research area, so one of my goals is to brush up on that.

May I ask what you plan to do with NNets? You must have a reason for studying them other then you see them as the future of AI.

Also what did CALC IV teach? Vector Calculus? stoke/divergence etc.

Neural networks are very brain-like and plausible. If a truly creative, humanlike AI is developed, I think it will probably look something like a neural network. No other AI technology seems very similar to how humans might think. I don't believe that we are subconscious logicians or run complicated algorithms to get things done. Our reflex responses only go through about 10 layers of neurons--no time for anything algorithmic. I think that if we can figure out how to represent and deal with abstraction in neural networks in a good way, we will have solved the basic problem of what human thought is.

Calc IV was vector calculus, but so was Calc III. There was very little that was new in Calc IV, just a few assorted applications like the TNB frame that were covered very shallowly.

If you can an imaging course might be useful as an app for NNets.

Lastly any course that will teach you Markov Chains(if you don't alreayd know it) would be a good course to take, if the prof isn't horrible or doesn't speak english properly.

Markov Chains are used in NNets and Reinforcement Learnig, a field related,if not contained in NNets.

I've had some exposure to Markov Chains in linear algebra and surprisingly also in my theory of computation class (probabilistic finite automata). It's probably worth thinking about.

 

[neurocomp2003]

If your interested in taking a look at Spiking Neurons or Reinforcement Learning. Look for these 2 researchers
Wulfram Gerstner(Spiking Neurons...he co-authored a book, don't rememebr the other author, LOTS of DEs)

Richard Sutton & ?? Barto (Reinforcemnet Learning)
Both Books tackle the concept of Adaptive Learning not seen in other NNet books.

Be wary that you will need to program your own testbed as these books are all theory. Rather intensive in Math and Algorithms. They are after all 4th year and Graduate Text.

Also you might want to take a look at the websites of people in the field(RL,SN,NNets,GAs)
and see if they have papers or references to papers they wrote and see if you can understand them. The AUthors mentioned above, Greg Hinton(and all his students), Sue Becker(student of Hinton), the group at CMU/UPITTs(Touretzky,Carson Chow, Bard Ermentrout), Simon Haykin,James Mclleland(can't remember how you spell his name), I think the name is Greg Anderson, C. Koch, The lady who wrote that very small GA book(melanie mitchell).

Also Gary Flakes book on Computational Beauty of Nature is a very nice read. and so is
Steve Grands book(neither has real math but interesting places to start)

The math that these people publish in their papers may give you and idea of what you are missing...For instant a lot of people publish about Gradient Descent/LSM...relatively simple concepts but I don't really know the math behind them myself(and the bayesian stuff,well the bayesian rule is simple enough)...thats why i really need to take a Mathematical Statistics course.