100 lines
4.1 KiB
Markdown
100 lines
4.1 KiB
Markdown
I have reviewed a number of papers relating to physics-informed machine
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learning principles.
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- Deep Tensor ADMM-Net for Snapshot Compressive Imaging
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- End-to-End Optimization of Optics and Image Processing
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- NeRF Basics
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- Implicit Surfaces via Volume Rendering
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- Continuum-aware NeRF (PAC-NeRF)
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- NeRF in Scattering Media
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- Lens Design with Differentiable Ray Tracing
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- Hybrid Lens Design with Differentiable Wave Optics
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- Diffractive Deep Neural Networks
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- Spatially Varying Nanophotonic Neural Networks
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- 3D Gaussian Splatting
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- Physics Integrated Gaussians
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- Intro to Graph Neural Networks
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- Interaction Networks for Learning Physics
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- GNNs as Learnable Physics Engines
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- Graph-based Physics Simulators
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- Deep Image Prior
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- GNNs and Generative Priors for Solving Inverse Problems
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- Invertible Generative Models
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- Diffusion Posterior Sampling
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Broadly, I have categorized these papers into four groups:
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- Data and Loss embedded physics. Physical constraints are applied in the
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training data and/or loss functions; any physical accuracy in the results is
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implicitly learned from these.
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- Architecture embedded physics. Physical constraints are applied in the model
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architecture. Typically this involves some sub-stage of the model which
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decodes a latent vector, applies some analytical physical formulae to it,
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then re-encodes it to a new latent vector for downstream processing. For
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example PAC-NeRF.
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- Operator embedded physics. The machine learning model does not directly
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produce outputs, but rather it produces some state representation which is
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then processed by an analytical physical model. For example differentiable
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implicit renderers as in NeRF or Gaussian Splatting.
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- System embedded physics. Some stage of the model involves a hardware physical
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step, such as optical neural networks or robotic feedback mechanisms.
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Synthesize a rough outline for a literature review paper which explores,
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generalizes, and unifies these four categories of Physics-Informed Neural
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Networks (PINNs).
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In particular, for each category, provide:
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- Brief introduction of the category
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- Unified formulation of architectures within the category
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- List of criteria to search for papers which describe models of the category
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- A coarse outline of that section of the review paper.
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We have already built some notions for unified formulation and outline for the
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first category, data- and loss-embedded physics, listed below.
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(note that we have not yet expressed robotic feedback mechanisms in this common
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formulation)
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---
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The driving formulation for all these models is that there is some general
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time-dependent nonlinear PDE of the form:
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d_t u(x, t) + N_u(x, t) = 0
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where N denotes a possibly nonlinear spatial differential operator, and u is
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the unknown physical field. The neural network u_theta approximates u, and from
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this we can solve inverse problems or forward inference.
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PINNs define a physics residual by substituting the neural approximation u_theta into the governing equation:
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r_theta(x, t) = d_t u_theta(x, t) + N_u_theta(x, t)
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The governing equation is satisfied at a point (x, t) when r_theta(x, t) = 0,
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so the physics loss penalizes based on this residual at various observations
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(x_j, t_j).
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That is, we optimize for the model parameters theta by
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argmin_theta | N_theta(x, t) - u_theta(x, t) |^2_2
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In architecture embedded physics, the model approximation `u_theta`
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incorporates some physics-based differential possibly nonlinear operator
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`partial` which informs the result.
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argmin_theta | N_theta(partial, x, t) - u_theta(partial, x, t) |^2_2
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In operator-embedded physics physics, the differential operator is applied to
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the model result.
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argmin_theta | partial(N_theta, x, t) - partial(u_theta, x, t) |^2_2
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Finally, in system-embedded physics, the model parameters and model are
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expressed in such a way that inference can be realized by a physical system.
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For example, of optical neural networks, the parameters theta are expressed in
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terms of diffraction gratings or nanophotonics. In this sense, the parameters
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are optimized by operator-embedded physics where partial is a free-space wave
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propagation operator.
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