Notes:
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Replace the synthetic DEM and rainfall with your real raster and event data.
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Add features (roughness, soil type, etc.) to node_static.
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Extend the decoder to predict multi-step depths or inundation probability maps.
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Add loss terms for physics-based regularization or temporal consistency.
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Tile the domain to reduce computational complexity
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Try the Muon optimizer.
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Use FMM to initialize a FNO (mesh independant so maybe no...)
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Compressive sensing to further sparsify the FMM solution
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Data analysis: come on dude.
Active Research: There is significant work, often termed Neural Dynamic Mode Decomposition (NDMD) or using DMD modes as an encoding for time, to integrate these concepts into deep learning models, including some that are operator-like. These methods aim to learn the optimal lifting function (encoder) and the linear Koopman operator (the kernel/transform) in an end-to-end manner.
These are not Fourier Neural Operators (FNOs) from Li et al., NeurIPS 2020. But Hydrologically Structured Neural Operator (HSNO) are conceptually related. I might try to use the FNO with the Fast Marching Method, but as FNO are mesh-independant, maybe that won't work. Here is a description of how the two approaches differ:
What are FNOs? FNOs were introduced as a way to learn mappings between infinite-dimensional function spaces, such as:
f:a(x)−>u(x)
where (a) could be coefficients or forcing fields of a PDE, and (u) its solution.
Mechanically:
They lift input fields to a latent representation.
Apply spectral convolution layers:
transform to Fourier space, multiply by learnable complex-valued weights on selected low-frequency modes, inverse transform back. Each layer performs a global mixing of spatial information (nonlocal receptive field) via these Fourier modes.
FNOs therefore approximate global integral operators — they are mesh-independent, fast, and data-efficient when PDE behavior is mostly smooth and translation-invariant (e.g., Navier–Stokes, Darcy flow).
What a HSNO does Your Hydrologically Structured Neural Operator (HSNO) differs fundamentally in both representation and inductive bias:
Aspect FNO (Li et al.) HSNO Domain representation Regular Euclidean grid (R²orR³) , FFT-friendly Irregular hydrologic graph derived from terrain (DEM) via Fast Marching Method (FMM) Operator kernel Learned in Fourier domain, globally shared Structured by travel-time topology, localized by flow connectivity Physics prior None explicit — translation-invariant kernel Explicit hydrologic physics via travel-time field, slope, DEM, and FMM graph Spatial coupling Dense, global (all points interact) Sparse, directional graph (upstream → downstream) Temporal modeling Often static or via separate time dimension Recurrent or sequence-based, with rainfall forcing and travel-time convolution Inductive bias Homogeneous isotropic PDEs Anisotropic flow along terrain gradients Convolution type Fourier spectral convolution (global) Travel-time convolution (causal, directed, physics-aware) So HSNO is not translation-invariant — it’s flow-invariant. It replaces the “global kernel in Fourier space” with a hydrologically structured kernel in travel-time space, respecting topography and flow direction.
Conceptual relation Formally, both FNO and HSNO can be seen as neural approximations of an operator ( G ):
u=G(a)
But the kernel parameterization differs:
FNO approximates
G(a)(x)=∫K(x−y)a(y),dy
where (K) is learned in the Fourier basis.
HSNO approximates
G(a)(x,t)=∫K(T(x,y)),a(y,t−T(x,y)),dy
where (T(x,y)) is the travel-time between points (from FMM) and (K) is learned in travel-time space rather than Euclidean distance. This makes HSNO causal and directional, which is essential for hydrology.
summary table Concept FNO HSNO Domain basis Fourier (frequency) Travel-time (hydrologic) Nonlocality Global Flow-directed Geometry Regular grid Graph topology (terrain) Operator type Stationary convolution Structured causal convolution Physical grounding Generic Hydrologic Why HSNO matters FNOs are elegant but agnostic to physical flow structure. In flooding, topography imposes strong directional anisotropy and time delays. HSNO internalizes that structure directly in the operator kernel — so instead of learning flow direction and delay from scratch, it starts from hydrologic reality and learns only the residual nonlinear dynamics (storage, infiltration, etc.).