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Bayesian Physics Informed Neural Networks for Data Assimilation and Spatio-Temporal Modelling of Wildfires

Joel Janek Dabrowski, Daniel Edward Pagendam, James Hilton, Conrad Sanderson, Daniel MacKinlay, Carolyn Huston, Andrew Bolt, Petra Kuhnert

arXiv:2212.00970v2cs.LG

TL;DR

Wildfire PINNs can fail to maintain temporal continuity under changes in external forcing, and level-set methods lack natural data assimilation and uncertainty quantification. This paper adds forecast and observation likelihoods to PINN optimisation and incorporates them into a Bayesian PINN. The authors report successful applications to two recorded grassland fires with data assimilation and uncertainty quantification, alongside results similar to the level-set method.

  • Problem

    Standard PINNs can lose temporal continuity under nonlinear changes in external forcing, while level-set methods lack natural data assimilation and uncertainty quantification.

    Source: §1 · p. 2

  • Method

    The paper proposes a forecast likelihood for temporal continuity and an observation likelihood for data assimilation, then incorporates these approaches into a Bayesian PINN.

    Source: §1 · p. 2, §5 · p. 30, §6 · p. 31

  • Results

    The PINN produced results similar to the level-set method, and the PINN and B-PINN were successfully applied to two recorded grassland fires with data assimilation and uncertainty quantification.

    Source: §1 · p. 2, §6 · p. 31

  • Takeaways & Limitations

    Bayesian PINN samples can provide plausible fire-front simulations for decision-makers and for post-fire reconstruction.

    Source: §5 · p. 30

    The mean-field assumption can underestimate true variance because it assumes independence across parameters.

    Source: §3.2.7 · p. 16

Abstract

from arXiv · show

We apply the Physics Informed Neural Network (PINN) to the problem of wildfire fire-front modelling. We use the PINN to solve the level-set equation, which is a partial differential equation that models a fire-front through the zero-level-set of a level-set function. The result is a PINN that simulates a fire-front as it propagates through the spatio-temporal domain. We show that popular optimisation cost functions used in the literature can result in PINNs that fail to maintain temporal continuity in modelled fire-fronts when there are extreme changes in exogenous forcing variables such as wind direction. We thus propose novel additions to the optimisation cost function that improves temporal continuity under these extreme changes. Furthermore, we develop an approach to perform data assimilation within the PINN such that the PINN predictions are drawn towards observations of the fire-front. Finally, we incorporate our novel approaches into a Bayesian PINN (B-PINN) to provide uncertainty quantification in the fire-front predictions. This is significant as the standard solver, the level-set method, does not naturally offer the capability for data assimilation and uncertainty quantification. Our results show that, with our novel approaches, the B-PINN can produce accurate predictions with high quality uncertainty quantification on real-world data.

1. Introduction

Wildfire modelling matters for emergency decisions under uncertainty, but level-set solvers lack natural data assimilation and uncertainty quantification. This study adapts PINNs to wildfire fire-front modelling and proposes ways to address temporal discontinuity and add assimilation and uncertainty quantification.

1. Introduction

  • The level-set method models wildfire fronts with a PDE but requires finely discretised grids, complicated derivative approximations, and reinitialisation to limit numerical errors.Its flexibility in following splits and mergers is an advantage.Source: §1 · p. 2
  • Level-set predictions lack natural data assimilation and uncertainty quantification, both relevant when fire-driving forces are uncertain.Without assimilation, inaccuracies can grow into large errors over time.Source: §1 · p. 2
  • PINNs solve PDEs with neural networks and offer continuous spatio-temporal predictions, data assimilation, and Bayesian uncertainty quantification.In wildfire modelling, the study identifies temporal continuity under changing forcings as a key challenge.Source: §1 · p. 2
  • The study proposes a continuity approach for nonlinear forcing changes, a PINN data-assimilation approach, and real-world demonstrations of assimilation and uncertainty quantification.The authors report PINN predictions similar to the level-set method, with the added capabilities of assimilation and uncertainty quantification.Source: §1 · p. 2

2. Related Work

Related work covers established wildfire modelling approaches and the trade-offs of level-set solvers, PINNs, and Bayesian uncertainty methods. Prior PINN studies leave temporal continuity and real-world wildfire assimilation and uncertainty quantification as open challenges.

2. Related Work

  • Environmental modelling work ranges from random forests and kriging to Bayesian hierarchical models and deep-learning emulation, including emulation of SPARK.Source: §2 · pp. 2–3
  • Level-set wildfire models use finite differences on fine grids, which can be computationally expensive and require reinitialisation to control numerical errors.Data assimilation and uncertainty quantification are not natural features of the level-set method; filtering-based additions can increase complexity.Source: §2 · p. 3
  • PINNs avoid numerical discretisation by using automatic differentiation, while Bayesian PINNs represent parameter uncertainty through posterior and predictive distributions.Variational inference is the most commonly used Bayesian PINN approach described here.Source: §2 · pp. 3–4
  • PINNs can be difficult to optimise, with long training times, limited accuracy, gradient pathologies, and spectral bias reported as challenges.Spectral bias can mean extensive training is needed to learn higher-frequency components.Source: §2 · p. 4
  • PINNs trained with common mean-squared-error losses can lose temporal continuity, particularly under abrupt forcing changes such as shifts in wind direction.Sequential training strategies do not enforce continuity; discrete PINNs may require retraining at each time step, while RNN approaches require unavailable time-step labels in wildfire applications.Source: §2 · p. 4
  • Earlier PINN applications to the level-set equation offered limited insight into obtained solutions or broader wildfire modelling challenges, and neither considered assimilation or uncertainty quantification.One implementation was constrained to a static fuel map.Source: §2 · pp. 4–5

3.1. The Level-Set Equation and the Level-Set Method

The level-set method represents a moving fire-front with the zero contour of a level-set function, evolves it under fire spread and wind, and uses discretisation and reinitialisation to compute stable trajectories.

3.1.1. The Level-Set Equation

  • The zero-level-set of u represents the fire-front, while the level-set equation evolves u through space and time under normal spread and external forcing.The resultant forcing vector combines endogenous and exogenous effects; the wind advection term is constrained to prevent backtracking when wind points against the normal.Source: §3.1.1 · pp. 5–6

3.1.2. The Level-Set Function and Distance Functions

  • The initial level-set function is positive outside the front, negative inside, and zero on its boundary; a signed-distance form is common, while an elliptical cone can better match elliptical fires.The elliptical cone is not guaranteed to satisfy the signed-distance condition ∥∇u∥=1.Source: §3.1.2 · pp. 6–7

3.1.3. Discretisation

  • The level-set method discretises space and time into a grid and advances the solution using a first-order hyperbolic upwind finite-difference approximation.Each update produces the level-set function at the next discrete time step.Source: §3.1.3 · pp. 7–8

3.1.4. Reinitialisation

  • Reinitialisation solves a PDE to restore ∥∇u∥=1 while preserving the initial zero-level-set, but numerical approximations can shift that front.The algorithm pauses time evolution periodically for reinitialisation, which is recommended sparingly.Source: §3.1.4 · pp. 8–9

3.1.5. Algorithm

  • The level-set algorithm initializes u, updates it at each grid time step, and periodically reinitialises it according to the chosen interval.Source: §3.1.5 · p. 9, §3.1.4 · p. 9

3.2. Physics Informed Neural Network

The PINN represents the level-set solution with a neural network and constructs the PDE through automatic differentiation. Its optimisation objective uses likelihoods for initial conditions, physics, forecasts, and observations, with unknown variances treated as weighting hyperparameters.

3.2.1. The PINN Architecture

  • A feed-forward neural network maps time, spatial coordinates, spread speed, and wind components to the level-set function; automatic differentiation supplies derivatives for constructing the PDE.The network represents a solution over a continuous spatio-temporal domain.Source: §3.2.1 · pp. 9–10, Figure 1 · p. 10

3.2.2. PINN Likelihood

  • The training data comprise initial conditions, PDE collocation points, Euler forecast values, and fire-front observations for assimilation.External conditions such as wind speed and direction are supplied as inputs.Source: §3.2.2 · p. 11
  • The B-PINN objective combines initial, physics, forecast, and observation likelihoods, with forecast and observation likelihoods introduced as novel additions.Unknown variances are treated as optimisation hyperparameters, making this approach closer to empirical Bayes than fully Bayesian inference.Source: §3.2.2 · pp. 11–12

3.2.3. Initial Likelihood

The initial likelihood fits the PINN to initial level-set values, while the physics likelihood uses the PDE to constrain its dynamics across space and time.

3.2.3. Initial Likelihood

  • The initial dataset Di supplies level-set values at sampled spatial locations to teach the network the initial shape of u(t, x, y).Locations may lie on a discrete grid or be sampled randomly; σi represents initial-condition error, including observation noise when the initial front is observed.Source: §3.2.3 · p. 12

3.2.4. Physics Likelihood

  • The physics likelihood penalizes the PDE residual at collocation points, constraining the network to follow the level-set equation without requiring level-set targets.The PDE itself functions as the target, making this a form of semi-supervised training.Source: §3.2.4 · pp. 12–13
  • Because the physics dataset is simulated, σp is tuned as a hyper-parameter rather than estimated from known data noise.An excessively small σp can make optimization favor the physics likelihood over other likelihoods; it is set larger than σi to balance PDE dynamics with fitting initial conditions.Source: §3.2.4 · p. 13

3.2.5. Forecast Likelihood and Temporal Continuity

The forecast likelihood addresses degenerate PINN solutions by comparing the current prediction with a forecast from the preceding time step. Its Euler-based forecast encourages temporal continuity, but can be inaccurate for highly nonlinear changes in time.

3.2.5. Forecast Likelihood and Temporal Continuity

  • Abrupt changes in the forcing variables can yield degenerate solutions, such as a PINN setting all derivatives to zero.The forecast-likelihood component penalizes these solutions and gives training an additional continuity constraint.Source: §3.2.5 · pp. 13–14
  • Comparing the current level-set prediction with an independent forecast from the previous time step adds temporal conditioning to encourage continuity.The forecast is based on an integral reformulation of the level-set equation and approximated using Euler integration.Source: §3.2.5 · p. 13
  • The forecast dataset uses increasing time steps of size Δt and compares each network output with the forecast from the preceding sample.Spatial locations remain fixed across time samples, and the dataset is generated on-the-fly during training; the PINN itself remains a continuous functional solution.Source: §3.2.5 · pp. 13–14
  • The Euler forecast is linear and can be inaccurate when u changes highly nonlinearly over time.Reducing Δt or using a higher-order method such as Runge-Kutta can reduce this error; forecast samples may focus on times or locations prone to degeneracy.Source: §3.2.5 · p. 14

3.2.6. Observation Likelihood and Data Assimilation

The observation likelihood assimilates fire-front locations by requiring the level-set function to be zero at observed points. It can use observations that cover only selected locations or times and may combine sources.

3.2.6. Observation Likelihood and Data Assimilation

  • Fire-front observations specify locations, times, and external conditions, and are treated as points on the zero-level-set where the level-set function should equal zero.Possible sources include firefighters and remote sensing such as satellite or drone imagery.Source: §3.2.6 · p. 14
  • Maximizing the observation likelihood penalizes solutions whose zero-level-set does not pass through the observed fire-front locations.Source: §3.2.6 · p. 15
  • Observations need not cover the full front or all times and may come from different sources, with σf chosen according to observation uncertainty and source.A sample may be a firefighter team's single point, followed by a set of satellite-derived points at the next time.Source: §3.2.6 · p. 15

3.2.7. Uncertainty Quantification and the Bayesian PINN

The Bayesian PINN approximates a posterior over network parameters using variational inference. The paper also names model configurations according to their likelihood components and whether they use Bayesian inference.

3.2.7. Uncertainty Quantification and the Bayesian PINN

  • The B-PINN approximates the parameter posterior with variational inference, optimizing variational parameters to minimize the negative evidence lower bound.The approach uses a mean-field posterior and an SSVS spike-and-slab prior.Source: §3.2.7 · p. 15
  • The mean-field assumption reduces computational complexity but can underestimate true variance because it assumes independence across parameters.Source: §3.2.7 · p. 16

3.2.8. Model Configurations and Nomenclature

  • PINN-e is the elementary configuration, PINN-f adds the forecast likelihood, and PINN-a adds the observation likelihood to the forecast configuration.B-PINN-a includes all likelihoods and B-PINN-f omits the observation likelihood; Bayesian configurations use variational inference, while the others use maximum likelihood optimization.Source: §3.2.8 · p. 16

3.2.9. Optimisation and Application

The paper describes gradient-based training for maximum-likelihood PINNs and Bayesian PINNs, then explains how Bayesian parameter inference yields predictive uncertainty.

3.2.9. Optimisation and Application

  • Training passes inputs through the network, evaluates outputs against targets with a cost function, and updates weights and biases using gradients over repeated epochs.The study uses either a maximum-likelihood or Bayesian cost function.Source: §3.2.9 · p. 16
  • Maximum-likelihood PINNs are trained by maximizing ln p(θ|D), while B-PINN parameters are optimized using the ELBO.For predictions, Monte Carlo samples from the variational posterior produce simulations whose distribution estimates the posterior predictive distribution.Source: §3.2.9 · pp. 16–17
  • Algorithm 2 applies maximum-likelihood training to PINN-a by evaluating initial-condition, physics, forecast, and observation likelihoods before updating parameters.The algorithm assumes time-ordered forecast samples; experiments use ADAM rather than gradient descent.Source: §3.2.9 · p. 17
  • Monte Carlo simulations from the Bayesian posterior predictive distribution support estimates of the prediction mean, standard deviation, and confidence intervals.The B-PINN optimizes variational parameters φ using the ELBO rather than updating θ according to the maximum-likelihood objective.Source: §3.2.9 · p. 17

4. Results

The experiments test forecast-likelihood PINNs on synthetic fires with changing winds and complex conditions, then evaluate assimilation and uncertainty quantification on two Australian fires. Results show improved continuity under wind shifts, closer data fit with assimilation, and B-PINN uncertainty coverage near the nominal target.

4.1.1. Synthetic Dataset

  • The synthetic fire changes from northerly to easterly wind after 10% of the simulation; Synthetic2 adds three obstacles, spatially varying speed, and time-varying wind.The first setup uses constant fire-front speed and an obstruction; Synthetic2 varies spread across space and wind over time.Source: §4.1.1 · p. 18, §4.1.2 · p. 18

4.1.3. Methodology

  • PINNs use grid-sampled initial, physics, forecast, and observation data; initial fire boundaries are oversampled, and evaluation uses Jaccard indices and posterior-interval coverage.Synthetic collocation points use a spatio-temporal grid; real-fire observations come from fire isochrones, and uncertainty coverage is evaluated against ground-truth fronts.Source: §4.1.3 · p. 19, §4.2.2 · pp. 24–25

4.1.4. Wind Direction Change Results

  • PINN-e changes to a simpler, physically unjustified level-set shape at the wind shift; forecast-likelihood PINN-f maintains continuity, with Jaccard index at 76% then above 80% otherwise.The wind changes direction at t = 0.1; the Jaccard index compares PINN predictions with the level-set method.Source: §4.1.4 · pp. 20–21, Figure 4 · p. 21, Figure 5 · p. 22

4.2.1. Fire Datasets

  • S03 and E06 represent variable-wind and dominant north-west-wind regimes; the real-fire evaluation uses manually mapped 10-second isochrones and measured wind inputs.S03's initial acceleration is not represented by the constant-speed model, and both approaches assume homogeneous fuel and wind conditions.Source: §4.2.1 · pp. 22–23, §4.2.3 · p. 25

4.1.5. Complex Fire-Front Shape Results

  • On Synthetic2, B-PINN-f represented complex fire-front shapes under spatially varying speed and wind, with 95% posterior predictive intervals narrower near obstacles.The simulation included three obstructions; the left region had higher speed, and the fire propagated faster in the predominant northerly wind direction.Source: §4.1.2 · p. 18, §4.1.5 · p. 22, Figure 6 · p. 23

4.2.4. Fire E06: Dominant Wind Fire Dynamics

  • For E06, data assimilation raised average Jaccard index from 0.72 to 0.84; B-PINN-a achieved 93.6% average uncertainty coverage against a 95% target.B-PINN-a and PINN-a had similar accuracy, while B-PINN-a additionally provided uncertainty quantification.Source: Figure 9 · p. 27, Figure 10 · p. 27, §4.2.4 · p. 28

4.2.3. Fire S03: Complicated Fire Dynamics

  • S03 B-PINN-a coverage averaged 89.4% against the 95% target, excluding the overestimated t = 30 result; the authors suggest mean-field assumptions may explain underestimation.They suggest reducing mean-field independence assumptions or using Hamilton Monte Carlo as possible improvements.Source: Figure 10 · p. 27, §4.2.3 · p. 28

5. Discussion

The discussion presents forecast likelihood as a response to PINN temporal-continuity failures under changing wind, alongside PINN-based data assimilation and Bayesian uncertainty quantification. It also describes inference trade-offs and boundaries for the approach.

5. Discussion

  • PINNs provide continuous spatio-temporal solutions and can assimilate data and quantify uncertainty, while level-set solvers rely on fine discretization and may require reinitialization.The PINN is described as naturally handling nonlinearity without discretization.Source: §5 · p. 28
  • Data assimilation can correct level-set propagation errors over time and support post-fire reconstruction by interpolating between observed isochrones.The observations may come from firefighters, drones, or remote sensing.Source: §5 · p. 28
  • The B-PINN provides uncertainty quantification and Monte Carlo fire-front simulations that can help decision-makers understand fire dynamics and support post-fire reconstruction.The level-set method does not naturally provide uncertainty quantification.Source: §5 · p. 30
  • Variational inference offers a faster alternative to MCMC for Bayesian neural networks, though mean-field assumptions limit posterior form and may also regularize it.MCMC can become prohibitively slow and expensive for large models.Source: §5 · p. 30
  • Spectral bias slowed learning of signed-distance discontinuities; oversampling near the zero-level-set, longer training, and dataset-specific activation choices addressed implementation challenges.The study used tanh for synthetic data and ReLU for fire data, with ReLU gradients potentially discontinuous.Source: §5 · p. 30
  • Forecast likelihood improved temporal continuity and allowed the PINN to model external wind-vector changes after sequence-to-sequence learning and weighted residual cost proved ineffective.The authors suggest the approach may also work in other applications.Source: §5 · p. 30
  • The proposed approach is not evaluated on mega-fires, which may involve heterogeneous environments, fire-generated winds, and spot fires moving against expected wind direction.These effects are not captured by simple models such as the Rothermel model.Source: §5 · p. 30

6. Conclusion

The conclusion presents forecast likelihood for temporal continuity, data assimilation, and Bayesian uncertainty quantification as contributions to PINN wildfire fire-front modeling. The approaches are demonstrated on two recorded grassland fires, and the authors argue that PINNs could be integrated into wildfire modeling platforms.

6. Conclusion

  • The paper addresses PINN temporal-continuity problems with forecast likelihood, develops PINN data assimilation, and demonstrates assimilation and uncertainty quantification on two recorded grassland fires.The forecast-likelihood approach is demonstrated on a synthetic dataset.Source: §6 · pp. 30–31
  • The authors argue that PINNs could be integrated into existing wildfire modeling platforms to facilitate data assimilation and uncertainty quantification.They describe the PINN as a practically useful solver for the level-set function.Source: §6 · p. 31
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