Spring 2026

prompt-00-outline.md

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