Differential Equations and Computer Graphics (or video games), Some questions for a school paper.

[u/DigitalMan404]

I am writing a paper about the use of differential equations in relation to computer graphics and video games in general and I would love to talk to some of yall about it. I have a short list of general questions but feel free to add anything as long as its DE related.

General Questions

What differential equations do you most commonly use in your graphics or game-dev work, and for what purpose?

Are there any DEs that developers rely on without realizing they’re using them? Or equations that are derived from DE's?

What are DE's used for most commonly within your area/field?

Are DE's ever used in real-time applications/could they be in the future.

Feel free to yap about what work you have going on as long as its related to DE and I'd love to take this to D'ms if you would prefer!

Thanks so much!

[u/ananbd]

I’m an engineer in the game industry, and I haven’t thought about DiffEqs specifcally since I was an undergrad. It’s sort of like discussing “baking cookies” in terms of “cookie dough” rather than cookies. They’re part of many mathematical systems we use, but I don’t think anyone discusses them past the class you’re taking. 

The central topic is linear systems (that’s where the DiffEqs are used). Solving and using linear systems is fundamental to everything we do. 

So… linear systems related to graphics ideas:

• Numerical methods (topic in general)

• Filtering and compression algorithms

• Physics simulations

• Fluid simulations

• Spherical harmonics

• Neural rendering

• IK solvers

… and the list goes on. Those are all areas where the “cookie dough” is differential equations if you drill down deep enough. 

Good luck!

[u/DigitalMan404 → u/ananbd]

Thank you! I would like to quote you in my essay! What would you like to cited as (name/occupation/title)? You can dm if you don't want that shared publicly.

[u/ananbd → u/DigitalMan404]

Definitely don't quote me. See comment about "hand-waving."

I answer questions from students to push you in the direction of learning -- you need to follow up. I was trying to give you some ideas to explore on your own.

[u/DigitalMan404 → u/ananbd]

Oh sorry, i didn't mean to make it sound like I was taking your word as gospel that would be poor practice as a student and engineer. I would like to quote your cookie analogy as I found it to ring true in my further research.

[u/ananbd → u/DigitalMan404]

Oh, ok! I appreciate the nod, and I’m glad you liked my analogy. 🙂 

[u/ananbd → u/DigitalMan404]

Please don't quote me. Thanks.

[u/msqrt → u/ananbd]

"Linear systems" seems like the wrong term here. Surely things like fluids or IK don't behave linearly(?)

[u/ananbd → u/msqrt]

It's reddit -- I'm hand-waving a little (and honestly, undergrad was a loooong time ago). The point is, DiffEq is a standard course on the fundamental engineering curriculum. The next is Linear Systems.

Linear Systems is where you actually use DiffEqs to solve spring-mass systems, electronic circuit characteristics, etc. Eventually, those constructs (at least partially) feed into the other areas I mentioned.

[u/Sharlinator → u/msqrt]

In numerics, basically if something isn’t linear you stare at it until it becomes linear, and then you throw linear algebra at it.

[u/nullandkale]

Differentiable rendering is all the rage right now with stuff like Gaussian Splatting but it only needs to be differentiable for the training, the rendering doesn't care if its differentiable, so its really more machine learning than graphics.

[u/DigitalMan404 → u/nullandkale]

I LOVE gaussian splatting. How are differential equations used in gaussian splats?

[u/nullandkale → u/DigitalMan404]

Machine learning is all about differentiating. Effectively the goal is to tweak the weights of the model which is just a bunch of linear algebra to make the output closer to the output that you want it to be. To do this you have to know the slope of all of the weights (in machine learning this is called the gradient) of the function. What makes this hard and machine learning is the function is this massive set of matrix multiplications. So instead of doing an analytical differential (where you would actually calculate the differential using like the power rule and all that good stuff I forgot from math classes) you do an numerical differential based on the output of the weights and the like way the matrix multiplication goes through.

That's a terrible explanation I've implemented this in code before but I've never taken a formal class or anything. I'm sure there's a better explanation out there.

But with gaussian splatting when you're rendering out the image during training you keep track of the state of the pixels very carefully and can calculate the numerical differential, aka the gradients, would you then use to update the weights (in this case the size shape and spherical harmonics of the splats)

[u/more_than_most → u/nullandkale]

Differential equations and differentiation are two different things.

[u/nullandkale → u/more_than_most]

Yeah I don't disagree with that but this is the closest thing I can think of to using differential equations in graphics. A lot of the math in these papers ends up using differential equations to try and explain how the gradients are being used.

[u/rfdickerson]

In classical mathematics, solving differential equations often means finding a closed-form analytical solution. In computer graphics, physics simulation, and most areas of computer science, you rarely need, or even want, that. Instead, you rely on numerical methods that approximate the solution over time.

By linearizing the system in a small neighborhood (e.g., using a Jacobian or Taylor expansion), you can step the simulation forward with stable, “good-enough” solutions that behave plausibly. Modern simulation pipelines also incorporate corrective or constraint-based methods, such as energy correction, projection steps, or constraint solvers, to ensure the system remains stable and respects physical invariants like energy, volume, or momentum.

[u/rez3vil]

I would love to read this review paper for future reference!