LEAP — A rapid neural surrogate of multi-fluid MHD at Europa
LEAP (Learning Europa's Atmosphere and Plasma) is a transformer-based surrogate of a multi-fluid magnetohydrodynamic (MHD) model of Jupiter's plasma interaction with Europa. It predicts the magnetic field perturbations along a spacecraft trajectory to within ±2.6 nT of the parent MHD, evaluating in under a second on a laptop rather than 12 hours on a 2,000-core cluster.
Overview
The Galileo mission measured magnetic perturbations near Europa's surface that were best explained by a global subsurface ocean. The strength of this oceanic signal is proportional to the salinity of the water which in turn tells about the types of life it could support. Characterizing Europa's sub-surface ocean is key objectives of the Europa Clipper and JUICE missions. The difficulty is that the measured field is a superposition of many fields: the Jovian background field, the spacecraft field, the ocean's induced response, and plasma interaction fields (PIFs) generated as Jupiter's co-rotating plasma flows past the moon. At times, PIFs can be comparable in magnitude to the oceanic field and vary rapidly, further complicating the measurement process.
Resolving the fields requires a physics-based models (e.g. MHD, magnetohydrodynamic). Though powerful, these models are computationally expensive, especially multi-fluid ones. The one used in this study has to compute the positions and behaviour of four couple fluids: electrons, magnetospheric O⁺, ionospheric/exospheric O⁺, and O₂⁺. A single run solves a ≈13.5-million-cell volume and takes more than 12 hours on a 2,000-core cluster. Historically this speed was considered to be annoying, but not prohibitive. However, the upcoming missions will return data every 2-3 weeks and the current physics- only models cannot keep up with the data volume, which represents a bottleneck in the ocean characterisation pipeline. To address this, we decided to see if a machine learning emulator (or copycat) could speed things up without compromising on accuracy.
The 'target' (thing to predict) is the plasma perturbation ΔB = BMHD − BJ, with the Jovian background removed (to improve ML stability). We add the Jupiter field back later using the Khurana Euler potential model.
Motivation
Europa Clipper's ECM investigation requires that the induced-field amplitudes at Jupiter's 11.2-hour synodic period and Europa's 85.2-hour orbital period are estimated to a precision of 1.5 nT. Meeting that precision means, in part, searching over the upstream plasma and atmospheric conditions, which are poorly constrained by observation.
Such a search requires many forward evaluations. A single MHD run occupies a 2,000-core cluster for upwards of 12 hours — of order 10⁴ core-hours — so systematic parameter surveys, and the inverse problem of estimating conditions from an observed field, are impractical with MHD alone.
The model
LEAP treats each flyby as an ordered sequence of time-aligned points. The self-attention mechanism inherent to transformers is useful here because the phenomena of interest are non-local: Alfvén wings and the global magnetic pile-up geometry are determined by the plasma interaction across the entire trajectory, not by conditions at a single point.
Architecture
LEAP is an encoder-only Transformer with L = 2 layers, H = 2 heads and d_model = 32, and an inner feed-forward dimensionality d_ff = 4096, deviating from the usual d_ff = 4·d_model convention. No positional encoding is used, since the spacecraft position (x, y, z, R) is included explicitly in the input vector. A linear head maps each hidden state to three outputs (ΔBx, ΔBy, ΔBz); |B| is derived rather than trained.
≈ 13.5 M cells
225× compression
d_ff = 4096
d_model = 32. Larger configurations (d_model 64/128/512, L and H of 3–4) showed declining performance, indicating minimal complexity is optimal for this domain.
Training data & the 24-rotation augmentation
Rather than learn the full MHD domain (300 × 150 × 300 ≈ 13.5 million cells), LEAP is trained on the ~60,000 points carved out along 63 real flyby trajectories from the Galileo (11), Juno (1), Europa Clipper (49) and JUICE (2) missions. We do this because, for this study, we mainly care about the magnetic field that the spacecraft will encounter, not the physics of the entire environment around Europa.
The resulting dataset is small, but the geometric invariance of MHD allows us to rotate each trajectory 24 times (number of symmetries of a cube), producing a factor x24 more data. Across 32 simulation volumes this yields up to 32 × 63 × 24 = 48,348 sequence samples, approximately 9 million learnable time-steps. Sequences are padded and masked to the same length within a batch (8) to improve training efficiency.
Physics-constrained loss
Beyond ~3 RE the environment is close to steady-state, so a vanilla L1 or L2 objective allocates capacity to the easy majority of points and gives higher error near Europa, where the field is most dynamic. The training objective is designed to counteract this.
The perturbation error is decomposed into components parallel and perpendicular to the local background field direction b̂, with the perpendicular term weighted more heavily (α = 0.5, β = 1.0) to emphasise transverse perturbations. A bounded radial weight wi = clip(1 − Ri/R_max, 0, 1) up-weights points close to the moon. A trajectory-based regularisation term, motivated by the ∇·B = 0 constraint, penalises rapid point-to-point variation (λ_grad = 0.1).
Training used Adam (η = 10⁻³, annealed ×0.8 every 8 epochs) with early stopping at patience 4, converging after 24 epochs in ~3 hours on a single A100. Five-fold cross-validation over unique trajectory IDs gives MAE = 1.76 ± 0.17 nT (9.6 %). A leakage assessment found no duplicate coordinates and a soft spatial leakage fraction of ~3.9 % within 102 km, indicating the test set is effectively independent.
Sweep the plasma yourself
The curves below are LEAP outputs for Europa Clipper flyby 15-E10 (closest approach 2031-10-23, 24 km altitude), computed across nine magnetospheric O⁺ mass densities from 320 to 2400 amu/cm⁻³ with the remaining inputs held fixed (H₀ = 5.0×10⁷, n₀ = 100, ocean response enabled). Peak perturbation amplitude increases monotonically with ρ₀, from 124.9 nT to 259.8 nT.
The δBz component captures magnetic pile-up on the trailing hemisphere and field-line stretching on the leading hemisphere, swinging from approximately −45 nT to +73 nT at the highest density. Each position of the slider corresponds to one forward evaluation of the model — the operation that makes the parameter surveys below tractable.
H₀ = 5.0×10⁷, n₀ = 100, ocean response on — only ρ₀ (O⁺ mass density) varies.
| ρ₀ (amu/cm⁻³) | peak |δB| (nT) |
|---|---|
| 320 | 124.9 |
| 580 | 146.7 |
| 840 | 169.0 |
| 1100 | 187.8 |
| 1360 | 205.9 |
| 1620 | 224.1 |
| 1880 | 238.6 |
| 2140 | 250.1 |
| 2400 | 259.8 |
Results
All components have a mean absolute error below 2.7 nT on the test set, with no obvious bias across the target range. Prediction error decreases with radial distance from Europa (mean error ±1 nT, remaining within 10 nT to the whiskers), consistent with the environment beyond 3 RE being less dynamic. For comparison, the ECM investigation requires the synodic and orbital induction amplitudes to be estimated to within 1.5 nT.
For the Galileo E4 and E14 flybys, held out in their native format, LEAP matches the parent MHD model in accuracy. Surveying 10,000 candidate plasma conditions per flyby and rejecting predictions where |B̂| exceeds 8.5 nT (E4) or 5.5 nT (E14) leaves 171 and 573 acceptable predictions, from which 5th and 95th percentile bounds and a median are derived. On E4 the survey yields statistical improvements in all three components, particularly By — significant because Europa's induction response is predominantly observed in Bx and By.
Parameter importance
Feature importance is assessed by ablating each physical parameter in turn and measuring the resulting MAE. Removing ρ₀ has the largest effect on Bx, increasing the error from 15.0 nT to 27.1 nT (+80.7 %), followed by n₀ (+47.5 %) and H₀ (+14.2 %). The error pattern is consistent across features, affecting Bx most, then Bz, then By. An earlier sensitivity analysis using the parent MHD code also identified ρ₀ and H₀ as primary drivers of the magnetic response in Bz. Geometric inputs (x, y, z, R) are excluded as they are not adjustable in the MHD model.
This ranking is a core proposition of an ML–MHD loop: it identifies where the model's sensitivity is highest, allowing expensive MHD runs to be prioritised in under-sampled regions of parameter space. Additional MHD outputs then improve subsequent versions of LEAP.
Parameter estimation
The acceleration allows the trained emulator to act as a near-instantaneous probabilistic forward model, so that constraining plasma conditions can be formulated as an inverse problem via simulation-based inference. The simulated field is modelled as normally distributed about the emulation, with σ² = 20 nT² per component and zero cross-covariance, and truncated-normal priors (all inputs are non-negative). The posterior over ρ₀, n₀ and H₀ is estimated using PyMC's slice sampler with four chains of ≥500 evaluations each, discarding 100 burn-in samples per chain.
Against input values ρ₀ = 320 amu/cm⁻³, n₀ = 25 (×10⁶ cm⁻³) and H₀ = 100 km, the E4 posteriors are ρ₀ = 252 ± 138, n₀ = 27.4 ± 4.2 and H₀ = 96.4 ± 5.9, with H₀ comparatively well constrained. E14 gives ρ₀ = 279 ± 179, n₀ = 33.6 ± 13.4 and H₀ = 85.4 ± 20.0. The broader n₀ posterior for E14 may reflect weaker constraint or enhanced variability in the trailing hemisphere plasma environment.
The inversion is performed on simulated MHD outputs, not spacecraft observations.
The inversion is performed on simulated MHD outputs rather than spacecraft observations. A direct inversion of observations would require taking the maximum posterior parameters inferred from the surrogate and re-running a forward MHD simulation to verify consistency; this end-to-end validation is not currently possible and remains future work.
Limitations
LEAP, like all surrogates, inherits the assumptions and omissions of its parent model.
- The BATSRUS model used here does not include electron-beam physics, revealed by Juno's 2022 flyby, nor the dawn–dusk asymmetry of Europa's O₂ exosphere represented in hybrid codes such as AIKEF.
- LEAP currently predicts only the magnetic field, unlike full fluid or hybrid models which output a suite of plasma quantities (density, pressure, velocity).
- All training simulations assume either no ocean induction (0 %) or a 100 % efficient induction response; no intermediate case is represented.
- The E14 prediction is capped at 13:27 UT, the boundary of the training volume (±2.5 RE in the y direction).
- Parameter estimation is demonstrated on simulated MHD outputs; it is not yet validated against in-situ magnetometer observations.
What's next
- A multi-modal approach incorporating both MHD outputs and in-situ magnetometry, to which the sequence-aware transformer is well suited.
- Extending the Bayesian inversion to in-situ observations, enabling real-time estimation of plasma conditions from ECM and PIMS.
- Companion models predicting a broader range of plasma parameters, which could serve as inputs for downstream test-particle calculations.
- Application to other bodies whose plasma environments remain poorly constrained, such as Uranus and Neptune.