Learning Physics You Can Trust: Seven Years in One Page
Everything I have written as first author is, in hindsight, one long argument: a neural network that predicts physics is useless unless you know when to believe it. The papers below are the steps of that argument, roughly in the order I stumbled through them.
Start: can a network see plasma at all? (2020)
Image Mapping the Temporal Evolution of Edge Characteristics in Tokamaks using Neural Networks was the naive question — can a branched convolutional network emulate SOLPS at the scrape-off layer, with coupled plasma and neutral behaviour, instead of running the solver? It could. Cheap surrogates for edge physics were on the table. But it was a black box mapping images to images, and nothing in it said how wrong it was.
Making the physics do some of the work (2023)
Two papers on making the learning problem easier rather than the network bigger. In Loss Landscape Engineering via Data Regulation on PINNs the finding was that PINNs fail less because of capacity and more because of a pathological loss surface — feeding in a regulated amount of data smoothens that surface and convergence follows. In Fourier-RNNs for Modelling Noisy Physics Data, a recurrent hidden state was bolted onto the Fourier operator, because real physics data is noisy and non-Markovian and one-step-in one-step-out is the wrong inductive bias for it.
Scale: the surrogate that actually ran on a tokamak (2024)
Plasma Surrogate Modelling using Fourier Neural Operators is the one that made the case at machine scale — a multi-variable FNO modelling plasma evolution inside a Tokamak, trained on JOREK simulations and then shown to hold on experimental MAST data. Six orders of magnitude faster than the solver. Which is exactly when the uncomfortable question arrives: this thing is fast and plausible, so how do I know its error bars?
The uncertainty years (2024–2025)
I did not have a good answer, so the next three papers are that answer.
Uncertainty Quantification of Surrogate Models using Conformal Prediction brings conformal prediction to spatio-temporal surrogates: guaranteed, distribution-free, model-agnostic error bars across the whole domain, with no assumptions about the network that produced them. Valid Error Bars for Neural Weather Models using Conformal Prediction carries the same machinery out of fusion and into neural weather models, because nothing in the argument was fusion-specific.
The catch in both is calibration data — you need held-out ground truth, which is the expensive thing you built the surrogate to avoid. Calibrated Physics-Informed Uncertainty Quantification closes that loop: use the PDE residual itself as the non-conformity score. The governing equation becomes the calibration set. Data-free conformal prediction, guarantees intact.
The review I wrote in 2025 on surrogate modelling and UQ of neural differential equations is the tidy version of that whole detour.
Now: build the structure into the model (2026)
The uncertainty papers bound the error from outside the model, after training. The two most recent papers put the structure inside the architecture, where training cannot remove it.
Learning Physical Operators using Neural Operators does this for the physics. Rather than learning the whole solution operator and bounding the damage afterwards, split the PDE: keep a fixed linear approximation of the operator you already know, and learn only the non-linear remainder as a neural operator inside a Neural ODE. Fewer parameters, better generalisation to physics never seen in training, and a model whose pieces mean something.
Learnable Splines for Unstructured Neural Operators does it for the geometry. The 2024 FNO worked because its data already sat on regular grids, but real engineering meshes are irregular, refined, and tens of millions of nodes large. L-SplineFNO moves the field between mesh and latent grid with a learnable B-spline transfer that keeps compact support, partition of unity and smoothness by construction. It scales to a 29.2-million-node model of the whole MAST-U tokamak on one GPU. Because the decoder is a spline, it also gives rigorous bounds on the prediction over continuous space and over boxes of uncertain parameters. The guarantee comes from the architecture rather than from a calibration set.
Arc, stated plainly: learn the map (2020) → help it learn (2023) → make it useful at scale (2024) → make it honest (2024–25) → build the physics and the guarantees into the architecture (2026). The trust problem was never solved by a bigger network. It was solved by giving the physics, and the mathematics underneath it, back their jobs.
Papers and code: Research · Google Scholar