Has the Deep Neural Network learned the Stochastic Process? An Evaluation Viewpoint
Authors: Harshit Kumar
Beomseok Kang
Biswadeep Chakraborty
Saibal Mukhopadhyay
Venue: ICLR 2025 Poster (2025)
Keywords: evaluation
deep neural network
stochasticity
complex systems
forecasting
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: A novel evaluation criterion to assess whether DNNs modeling stochastic complex systems have learnt the underlying stochastic process

Abstract: This paper presents the first systematic study of evaluating Deep Neural Networks (DNNs) designed to forecast the evolution of stochastic complex systems. We show that traditional evaluation methods like threshold-based classification metrics and error-based scoring rules assess a DNN's ability to replicate the observed ground truth but fail to measure the DNN's learning of the underlying stochastic process. To address this gap, we propose a new evaluation criteria called _Fidelity to Stochastic Process (F2SP)_, representing the DNN's ability to predict the system property _Statistic-GT_—the ground truth of the stochastic process—and introduce an evaluation metric that exclusively assesses F2SP. We formalize F2SP within a stochastic framework and establish criteria for validly measuring it. We formally show that Expected Calibration Error (ECE) satisfies the necessary condition for testing F2SP, unlike traditional evaluation methods. Empirical experiments on synthetic datasets, including wildfire, host-pathogen, and stock market models, demonstrate that ECE uniquely captures F2SP. We further extend our study to real-world wildfire data, highlighting the limitations of conventional evaluation and discuss the practical utility of incorporating F2SP into model assessment. This work offers a new perspective on evaluating DNNs modeling complex systems by emphasizing the importance of capturing underlying the stochastic process.

Published as a conference paper at ICLR 2025
HAS THE DEEP NEURAL NETWORK LEARNED THE STOCHASTIC PROCESS? AN EVALUATION VIEWPOINT
Harshit Kumar, Beomseok Kang, Biswadeep Chakraborty & &Saial Mukhopadhyay School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Georgia, USA
(hkumar smakhopadhyay6 egatech.edl
ABSTRACT
This paper presents the first systematic study of evaluating Deep Neural Networks (DNNs) designed forecast the evolution of stochastic complex systems. We show that traditional evaluation methods like threshold-based classification metrics and error-based scoring rules assess a DNN's ability to replicate the observed ground truth but fail to measure the DNN's learning of the underlying stochastic process. To address this gap. we propose new evaluation criterion called Fidelity Stochastic Process (F2SP). representing the DNN's ability to predict the system property Statistic- -Tetie ground truth of the stochastic process -nd introduce evaluation metric that exclusively assesses F2SP. We formalize F2SP within stochastic framework and establish criteria for validly measuring it. We formally show that Expected Calibration Error (ECE) satisfies the necessary condition for testing F2SP, unlike traditional evaluation methods. Empirical experiments on synthetic datasets, including wildfire, host-pathogen, and stock market models, demonstrate that ECE uniquely captures F2SP. We further extend our study to real-world wildfire data, highlighting the limitations of conventional evaluation and discuss the practical utility of incorporating F2SP into model assessment. This work offers a new perspective on evaluating DNNs modeling complex systems by emphasizing the importance of capturing the underlying stochastic process
INTRODUCTION
Deep Neural Networks (DNNs) increasingly employed model complex systems and forecast their evolution across diverse fields, including infectious disease spread (Keshavamurthy et al. 2022]
Ibrahim et al. 2021 Bomfim al. 2020 Kuo & Fu. 2021). finance (Li et al. (022)), weather
prediction (Scher & Messori 2021
Bonavita & Laloyaux] 2020), geophysics Yu & Ma 2021
Lasistro-Hard et 20221), and wildfire prediction (Huot al. [2022] Radke al., 2019; Yang et al. 2021). These systems exhibit complex global dynamics arising from relatively simple, localized, and stochastic interactions between participating elements, which are often referred to as agents from satellite imagery to population records DNNs aim to learn the agent interaction rules for long-horizon forecasting.
(Layyman et al. 2013). Leveraging large-scale, multi-dimensionn
A fundamental characteristic of such physical systems is randomness, making their evolution inher- ently stochastic; thus, identical initial conditions can lead to multiple outcomes, forming a stochastic process. However, we typically observe only single outcome of the system's 's evolution, a system property that we term the Observed Ground Truth (Observed-GT), Current evaluation metrics primarily assess how closely a DNN's predictions match this Observed-GT, an evaluation criteria call Fidelity to Realization (F2R). This focus on F2R raises a critical question when DNN fails to match the Observed-GT: is the mismatch due to inherent stochastic variability, does it stem from exposure to a fundamentally different stochastic process the DNN has not modeled? Understanding this distinction is key: DNN that captures the stochastic process but mismatches the Observed-GT may still be valuable, while failure to model the stochastic process entirely undermines its utility.
Coode (Code-httpps:/ / //gi wtt.gittbb..oo. httpp........a.atttikka///c///// htwp://.cnn/aaaaaiitkeee//lltaee s_ppocos

---

Published as a conference paper at ICLR 2025
DNN Prediction
Proposed Evaluation Framework
time
Observed
F2SP
ECE
MSE
Observed-GT
Statistic-GT
AUC-PR
F2R
Stochastic Process
Evaluation
System
Criteria
Property
Staledi
GT
F2R
Observed-GT
F2SP
StatisticGGT
(a)
(b)
Figure 1: (a) The figure depicts the evolution of a stochastic process forest fire model. Starting from same initial conditions, diverse outcomes emerge over the prediction horizon, depicted by the shaded red region. The Observed-GT (br,(i.5)) H represents one outcome on a x W grid, while the Statistic-GT (Pt.(t.1) x shows the normalized frequency of target state occurrences, capturing the full stochastic process ($ (b) This panel illustrates the proposed evaluation framework: F2R evaluates alignment with Observed-Gi (using AUC-PR), F2SP tests alignment with Statistic- GT (using ECE), and MSE balances both criteria. The framework provides unified approach for interpreting model performance in stochastic settings. See { for practical guide on the framework.
To address this problem, we formulate a new system property called the Statistic Ground Truth (Statistic-GT), Statistic-GT captures the system's 's expected behavior across all possible outcomes from the same initial conditions. It provides complete representation of the stochastic process and serves as stable target for evaluation (Figure aa). With this new target, we propose an evaluation criterion called Fidelity to Stochastic Process (F2SP). which measures how faithfully the DNN predicts the Statistic-GiT. A key challenge arises from the fact that Statistic-G" is not directly observable in practical scenarios; only the Observed-GT is available, providing a partial glimpse of the complete stochastic process. Thus, the challenge is testing F2SP using only the Observed-GT.
To address this challenge, establish that Expected Calibration Error (ECE) is suitable evaluation metric that satisfies the necessary condition to test F2SP, whereas commonly used metrics like classification-based (e.g., Area Under the Precision Recall Curve, AUC-PR) and probabilistic error- based metrics (e.g., Mcan Squared Error, MSE) fail to do so (Figure bb. $ We conduct benchmark experiments on three synthetic complex systems- forest fire (Hargrove
2000). host-pathogen (Sayama 2013), and stock market (Wei et al. 2003) ($2) demonstrating ECE's unique behavior across varying levels of stochasticity ($4). Finally, we evaluate on the real- weal-world "Next Day Wildfire Spread" dataset (Huot et al.) (022). showcasing the limitations of F2R strategy and the practical utility testing F2SP ($5). While ECE is traditionally used estimate calibration error, our work discovers ECE's s new utility for testing F2SP, broadening its role ($6). Our key contributions are:
We identify critical limitations current evaluation strategies focused solely on matching the Observed-GT, highlighting the need for more robust approach in stochastic systems.
We propose F2SP to assess DNN predictions against Statistic-Gr GT, the system's 's expected behavior across all outcomes, ensuring stable evaluation in stochastic scenarios.
We demonstrate, both formally and through benchmark experiments, that ECE uniquely satisfies the necessary condition for testing F2SP using only the Observed-GT.
Beyond synthetic systems, analyze real-world wildfire data, identifying instances where stochasticity disrupts traditional metrics and observing trends that align with our synthetic findings, reinforeing the practical applicability of our study.
By shifting the focus from replicating Observed-GT to capturing underlying stochastic dynamics, our work provides novel perspective on evaluating DNNs that model complex systems. We recommend adopting the proposed evaluation framework (shown in Figure [b), which integrates ECE for assessing F2SP alongside F2R metrics, such classification- based and proper scoring rules, to ensure a stochasticity crocasticit--comptbbbe evaluation strategy. This simple framework has significant implications for improving model evaluation, which this paper aims explore and highlight.
2

---

Published as a conference paper at ICLR 2025
2 BACKGROUND AND DATASET
This section formalizes the DNN prediction and evaluation framework ($2.T) It describes real-world challenges in observing the effects of stochasticity, motivating the use of synthetic benchmarks 222.) Finally, it covers synthetic systems used in this work and their stochastic simulation
2.1 FORMULATION OF DNN PREDICTION AND EVALUATION FOR COMPLEX SYSTEMS.
We model the evolution of a complex system on a grid of size H x W, where each grid cell (i, j) at time & can occupy one of m discrete states, denoted as St,(i,)) 82,,8......
We are
particularly interested in tracking specific state s". Let (t,)) E {0, represent whether state s* is present (1) or absent (0) in cell (i,j) at time f. collectively forming the grid-level ground truth B, (bt,(t,)) | Additionally, O E (R"))/*W" represents an n-dimensional vector of observational variables (e.g.. environmental factors, agent attributes) for each of the H x W grid cells at time The DNN predicts the probability of state s" being present at each cell (i,j), denoted by Pr.( These predictions form the joint conditional distribution P. = (f.,(t.)) } " predicted T timesteps into the future, using past states B1:--T and observational variables O1a-7:
P = P(B, BI.-T: O14-T)
(1)
After prediction, we use an evaluation metric S(B., P.) measure prediction accuracy. Different choices of S assess different system properties (see $3.4) In Appendix B Table 3) serves reference for the notations used throughout the paper.
(10, 1] x 10x1) + I to
2.2 STOCHASTICITY IN COMPLEX SYSTEMS AND CHALLENGES IN STOCHASTIC MODELING
Stochasticity in complex systems stems from randomness in interactions driven by environmental factors, agent behaviors, and external interventions, adding noise the "evolution rules" DNNs aim to Iearn. For example, the spread of wildfires is shaped by vegetation, terrain, weather, and human activity (Liu et al.) 202T), while infectious disease dynamics depend on movement, social interactions, and interventions (GroBmann et al. 2020). Small variations in these interactions can cause divergent outcomes, similar to the "butterfly effect" (Lorenz) 2000), complicating long-term forecasting.
Real-world data, providing only a single observed outcome, obscures the true implications of stochastic interactions in complex systems. To circumvent this limitation, we use simulation-based approaches like agent-based models (ABMs) (e.g.. NetLogo (Tisue & Wilensky (0044), proven effective in capturing real-world complexities (Manson et al.) (0122). See $C.1 for applications of ABMs in complex system modeling. In this study, ABMs generate range of potential evolutions based on stochastic rules, allowing us to study emergent behaviors beyond real-world data constraints.
2.3 SYNTHETIC COMPLEX SYSTEMS AND THEIR STOCHASTIC SIMULATION
Synthetic environments in this work We use three synthetic environments forest fire (Hargrove
et al. 2000), host-pathogen (Sayama 2013), and stock market models (Wei et al. 2003) as examples
Table 1: Overview of the synthetic complex systems used in this study.
Environment
Competitive
States
Interaction Rules
Description
Forest Fire
No
empty, patch, fire, ember (uiee, ember]
Fire spreads to neighboring patches with probability Piyne- based Rothermel's is heat transfer model.
Simulates fire spread through land patches without opposing Agents spread fire with- out competition between states.
Pathogen
empty, dead, healthy, infected (healthy)
Healthy agents are infected probability Patess infected agents die with Pirat, and dead cells are cured healthy cells with Pewe-
Models disease transmission and recovery, with competing healthy and infected states.
Stock Market
Yes
hold, sell, buy, inactive (bay)
Investors buy, sell, or hold based on market sentiment neighbor ac- tions with probability Proce-
Simulates competing investor ac- tions (buy vs. sell), influencing market trends.
3

---

Published as a conference paper at ICLR 2025
of complex systems. While all three models consist of four discrete states $t,(k.5) {s1, 82, 83, sal. they have completely different interaction dynamics. Further, the Host-Pathogen and Stock Market models feature competing agents, reflecting socio-cconomic dynamics, unlike the natural processes in the Forest Fire Model. We assume homogeneous agents (no observational variables Oz.(1.)) to focus solety stochastic dynamics. See Table I overview and $D.1 for detailed descriptions.
Agent-Based Simulation Framework. We simulate on 64 x 64 grid, where each cell (i.j) at time t represents an agent state $1,(1.5) Simulations start with agents randomly assigned initial states $0.(1.) based on distribution P (()o.(t.))) Agents interact with their neighbors according to stochastic transition rules (6))) (st, Mi,,)) where M(.i) represents the agent's Moore neighborhood. These local interactions drive global behaviors, manifesting as complex, fractal- like patterns. The stochasticity in the interaction rules introduces randomness into the system's evolution, simulating the diverse behaviors. To control stochasticity, we use the S-Level parameter, with higher S-Levels introducing more randomness and S-Level O representing deterministic interactions.
Role of S-Level and its realism. The S-Level acts control knob for adjusting stochasticity. Its focus is not on realism, but on precise control of the stochastic process. This enables robust assessment of evaluation metrics' fidelity to the system property Statistic-GT across varying levels of randomness (see (3.2). In real-world datasets ($5). the concept of S-Level is more complex, stochasticity be represented using distribution of S-Levels shaped by environmental and temporal dynamics. However, the fundamental property of Statistic-GT remains relevant.
EVALUATING A COMPLEX STOCHASTIC PROCESS
This section formalizes Statistic-GT as an evaluation target aligned with system stochasticity ($3. 3.2 and emphasizes its importance by highlighting the limitations of F2R strategy ($3.3). Finally, provides theoretical insights showing why ECE uniquely tests F2SP using Observed-Gi ($3.4).
3.1 SIMULATING A COMPLEX STOCHASTIC PROCESS
Empirical Stochastic Process (ESP). An ESP is represented through Monte Carlo (MC) simulations (1000 in study), each starting from identical initial conditions and rumning until all state transitions cease. Identical initial conditions ensure variations in fire evolution arise solely from stochastic interactions. In the wildfire example, the forest configuration and fire seed location define the initial conditions, and the simulation runs until fuel depletion. Figure 2 shows four distinct MC simulations (MC-1 to MC-4), illustrating different evolution pathways from the same starting point.
time
MC-1
Simulation Strategy
Setup
Deterministic Train/Test
Stochastic TrainyTest
MC-2
ESP
Initial Conditions
Forest Configuration
X
MC-3
Fire Seed
Location
X
Statietic GT
S-Level
0
0
Rendom
X; Foved
Figure 2: (a) The first four rows display four distinct MC simulations of forest-fire evolution from the same initial condition. The last row shows the Statistic-GT representing the evolution of the Stochastic Process (formally defined in $3.2h). For qualitative examples of MC simulations from other complex systems, please refer to ID (b) Table highlights the sources of randomness in the synthetic dataset across different simulation strategies used this study. Deterministic processes randomize initial conditions but follow fixed fire evolution rules (S-Level 0). Stochastic processes (S-Level > 0) allow multiple evolutionary paths: the ESP fixes initial conditions, while stochastic train/test setups (used in benchmark experiments, randomize initial conditions across simulations.
4

---

Published as a conference paper at ICLR 2025
Random instances of deterministic vs. stochastic evolution. Figure 2 highlights the sources of randommess in the forest fire dataset: forest configuration, fire seed location, and fire evolution rules. In deterministic evolution, only forest configuration and fire seed location are randomized, while fire follows a single pathway, producing random instances of deterministic fire. In contrast, the ESP allows multiple evolutionary paths from the same initial conditions due to stochastic evolution. When all elements are randomized, we obtain random instances of stochastic fire evolution, used to train DNNs in 44.1 This synthetic dataset simulates simplified version of real-world scenarios.
3.2 FORMULATING COMPLEX STOCHASTIC PROCESS
Micro-Level Modeling. At the micro-level, each grid cell (i:j) at time f is modeled using a binary random variable (RV) (..), which captures the stochastic nature of the GT state (1,)).)- M,,6.)) follows a Bernoulli distribution with probability parameter Pt,(k.)- This parameter is derived from the ESP by normalizing the frequency of s* across all MC simulations at each timestep.
Statistic-GT. While individual Micro RVs, Mt.(..)) represent independent probability parameters, } J captures the joint probability distribution across the entire grid
the ensemble P. {pcc(e.p)) as defined in Equation [T] These grid cells are spatially and temporally interdependent, forming the Statistic-GT. This emergent behavior of the micro-level statistic (t,(i.)) is representation of the full stochastic process. The evolution of Statistic-Gi is depicted in the fifth row of Figure 2 Typically, only single instance from this stochastic process (one MC simulation) is observable practice.
3.3 WHY IS STATISTIC-GT PROPERTY OF INTEREST: LIMITATIONS OF F2R
In complex systems, evaluation metrics assess a DNN's performance by aggregating micro-level comparisons between the ground truth br,(k,) and predicted forecast (t,(..)) for each grid cell (i.j). This aggregation yields a macro-level score that summarizes model performance across the grid, reflecting its ability to replicate the Observed-GT. However, in stochastic systems, reliance observed instance of the system's 's evolution increases sensitivity to variability in the Observed-GT. This sensitivity creates challenges for the F2R strategy. We analyze this sensitivity by examining how variability in Observed-GT affects the stability of evaluation metrics, including classification- based metrics, MSE, and ECE, as detailed in F.I with key findings summarized in the main paper.
single
To formalize this analysis, we define the Macro RV Zt, which represents the total number of grid cells in the target state s* at time This variable provides a grid-level summary of the system's evolution, analogous to the calculation of the evaluation metric. Next, we show that classification- based metrics are particularly sensitive to macro- varianee (V ar (2:) because they exclusively test F2R. Higher Var Zz] increases mismatches between Observed- G and DNN predictions, leading to: (1) score degradation due randomness and (2) metric fluctuations caused by GT variability, falsely indicating model instability. MSE and ECE are less sensitive to V VarlZi because they do not exclusively test F2R. This observation underscores the instability of Observed-GT as a system property for stochastic systems, making F2R inadequate for evaluating models in highly stochastic scenarios.
In contrast, the Statistic-Gi represents the system's 's expected behavior across all possible outcomes, offering more stable evaluation target. Testing F2SP evaluates whether the model has learned the underlying stochastic process, ensuring robust performance assessment in highly stochastic systems.
3.4 HOW DO WE TEST FIDELITY TO STATISTIC-GT?
As stated in Equation given DNN's prediction P; and the Observed-GT map Br, the evaluation metric S(B,, Pt) measures the fidelity of the DNN's predictions Our focus is on the property Statistic-GT denoted by P.. The key challenge is testing a DNN's fidelity Statistic-GT using only Observed-GT, Statistic- GT is not directly measurable.
specific property of the system.
In our research, we discovered that ECE, an evaluation metric commonly used in Deep Learning and calibration error estimation, possesses the unique ability to test fidelity to Statistic-GG GT using only the Observed-GF GT. Next, we formally show how ECE achieves this and compare it to standard baseline metrics, that lack this capability. In $4 we empirically validate this discovery.
5

---

Published as a conference paper at ICLR 2025
3.4.1 EXPECTED CALIBRATION ERROR TESTS FIDELITY TO STATISTIC-GT
We formally show that ECE satisfies the necessary condition for evaluating DNN's fidelity to the Statistic-GT. Consider a grid of size H x W at time 6, where each cell (i, (, is modeled as a Bernoulli random variable Me,(3.)) ~ (ern(pt,(t.))) indicating the presence (1) or absence (0) of 8* The DNN
outputs predicted probabilities (t,(i.)) for each cell, forming the set P. { ) |BXW,
We group the predicted probabilities into K bins 17- = (+7,, x and define H₁. = {(i,j) (1,0.)) } as the set of indices in bin 4.- Let (i.j).) E {0, be the Observed-GT at cell (i.j). For each bin [H& E(,j)E bt.(t.)). A calibration curve plots frac(k) against the representative predicted probability P& (usually the bin midpoint). perfectly calibrated model satisfies frac(k) P& for all bins. The Expected Calibration Error (ECE) (Naeini al. 2015 summarizes calibration as: ECE
4. the fraction of positives is: frac(k)
E [Hul [frac(k)) p-ll
ECE tests fidelity to Statistic-GT. For perfect predictor where Pt.(k.s) Pt.(i)) for all cells, we examine E [[frac(k]]. Since (i,() are realizations of Mr,(ss), we have B(br,(t..). Ptçii)) (r.(i.)) Therefore.
[k]raa(A)]]
1
Σ (K.))Hz
Elbb()))
Σ
PL, pu.
|/
HP
(i.))
This shows that, on average, frac(k) = p for a perfect predictor, implying zero ECE (see $.22 for calibration curves). Intuitively, calculating frac(k) marginalizes over data points in bin Lg. treating them as independent and ignoring dependencies among (k,( for all (i, E Hg- Thus, a low ECE satisfies the necessary condition for evaluating fidelity Statistic- but not the sufficient criterion.
Application of ECE in Deep Learning vs. Our Work. While ECE is commonly used to assess the calibration of DNN predictions in tasks like image and text classification, investigates its properties for evaluating complex system forecasting. Unlike prior studies that simply ECE for measuring DNN output calibration, to fundamental system property (Statistic-GT) that aligns with the stochastic nature of these systems. By addressing system randomness through F2SP, we position ECE as a complementary metric. Our study establishes perfect calibration as a necessary condition for testing F2SP, fundamental insight overlooked in prior works, emphasizing that calibration should be central to the evaluation of such stochastic systems rather than secondary to discriminative performance (see C.2 for details).
work uniquely
identify its unique suitability for assessing fidelity
3.4.2 BASELINE EVALUATION METRICS
Classification- Metrics (Ferri et al., 2009). The prediction problem can be framed as a classification task, where the goal is to predict whether each grid cell will be in specific state (e.g.. presence of s*), Metrics like Precision, Recall, Fl-score, and AUC-ROC/PR are commonly used. These metrics, based on thresholding ranking, are favored for distinguishing between Type and Type II errors. They exclusively assess how well thresholded predictions match the Observed-
Proper Scoring Rules. Scoring rules evaluate probability forecasts by ensuring the best score is achieved when the forecast matches the GT distribution (Gneiting Raftery 2007). Strictly proper scoring rules, such as MSE, Binary Cross-Entropy (BCE), and Continuous Ranked Probability Score (CRPS), are commonly used in forecasting. These rules promote sharpness by penalizing uncertain predictions and can be decomposed into calibration and sharpness components (Gneiting & Raftery 2007: Ramos et al. 2018). For example, MSE can be decomposed using the Brier Score decomposition (Blattenberger & Lad 1985):
M Bml N
M
Buul
MSE-
(frac Bm) Pmn) ) +++ N
frac Bm (1 frac (Bm)),
m=1
Calibration
Refinement
where the Refinement term promotes sharpness by penalizing uncertainty. This decomposition highlights a key distinction: scoring rules penalize uncertain predictions at the micro level, making them directly influenced by Var[Z,], whereas ECE focuses solely on calibration and remains unaffected by Var [Z,] in its convergence. We elaborate further this distinction in $D.

---

Published as a conference paper at ICLR 2025
4 BENCHMARK EXPERIMENTS
4.1 EXPERIMENTAL SETUP
We employ a single-layer ConvLSTM architecture for its simplicity (Shi et al., 2015). designing the ConvLSTMM-CA variant preserve spatial information essential for learning cellular automata interaction rules ($E.I). Our experiments span various S-Levels, with 1,000 simulations for each S-Level: 0, 5, 10. 15, and 20 for the forest fire model: 10, 15, and 20 for the host-pathogen model; and 5, 10, and 15 for the stock market model (see $D. S-Level definitions). Simulations are initiated with randomized conditions and configurations, representing distinct random realizations of specific stochastic processes, with varying S-Levels indicating different process complexities.
Training and testing parameters include the observation period fats and the prediction period fpredt The forest fire simulations use lass 10 frames and (pred 50 frames, whereas the host-pathogen and stock market models use lohs- 10 and pred = 20 frames. For each S-Level, 700 simulations are used for training and 300 for testing. The DNN predicts the subsequent Epred frames after observing the first tobs frames. We employ BCE loss for model training.
While the main paper presents ConYLSTM-CA results for AUC-PR and MSE, extended evaluations across multiple DNN architectures are detailed in SF3 This includes multi-layer, spatial bottleneck ConvLSTM variants (Shi al.) 2015) and Attentive Recurrent Neural Cellular Automata (AR-NCA)
(Kang et al. 2024), which specializes in modeling locally interacting discrete dynamical systems.
4.2
TESTING ECE'S ABILITY TO ASSESS DNN'S FIDELITY TO STATISTIC-GT
1 AUC-PR
MSE
ECE
0.32
0.11
0.18
20: 0.007 0.014 0.034
0.064
0.115
20
0.099
0.085
0.073
0.059
0.051
20
0.156
133
0.098
0.052
0.004
Level
15-0.003
0.010 0.029
0.859
0.112
0.21
0.053
0.045
0.044
0.047
0.059
0.12
0.099
0.064
0.003
0.047
0.12
10.
0.067
0.11
5.0.011
0.073
0.116
0.06
5.0.023
0.074
0.125
0.06
0-0.000 0.029 9.094
188
3.317
0.0.004
6. 0.026
0. 0.066
0.124
0.173
0.0.002
0.025
0.067
336
0.176
0.00
0.00
0.000
0.37
0.12
0.09
20-0.243 0.201
0.255
20-0.045
0.057
0.085
20
0.066
0.031
0.003
0.31
0.09
0.05
15. 0.193 0.177
0.293
15 0.029 0.049
0.096
15
0.036
0.003
0.042
0.28
0.05
0.03
10.0.176 0.208
0.370
10.0.020
0.061
0.120
10.0.001
0.049
0.095
0.59
0.10
15.0.445
0.572
0.556
15. 05.0.3
0.062
0.092
0.065
0.002
0.05
0.424
0.569
0.558
0.032
0.03
5.0.409
0.585
0.583
0.001
0.036
0.075
0.03
0.00
Test B-Level
Test S-Level
Test S-Level
Figure 3: Performance of DNNs trained on one S-Level and tested on another, evaluated using three evaluation metrics: (a) (1- AUC-PR) J. (b) MSE J. and (c) ECE I across three complex systems. Top Row: Forest Fire, Middle Row: Host- Host-Pathogen, Bottom Row: Stock Market. The z-axis (Test S-Level) and y-axis (Train S-Level) are consistent across all matrices. This figure highlights ECE's unique ability to evaluate whether the DNN has learned the correct stochastic process. While AUC-PR and MSE exhibit performance degradation as the difference between train and test S-Levels increases, ECE shows distinct pattern. Specifically, for cases where the train and test S-Levels match, ECE indicates little to no performance degradation, underscoring its unique capability to test the F2SP evaluation criteria. Theoretical insights into these results detailed in $ and 3.4.2
7

---

Published as a conference paper at ICLR 2025
4.2.1 SENSITIVITY TO S-LEVEL: ECE vs. BASELINES.
In $ E..2 we confirm that S-Level determines the Statistic-GTTs appearance. We also verify that the DNN, trained with the self-supervised approach from $4.1 has learned to predict the Statistic- Thus, an evaluation metric that measures DNN's fidelity to Statistic- should exhibit sensitivity to the match between training and test S-Levels. To validate this, we create a matrix heatmap that evaluates DNNs (in $4.1 each trained specific S-Level, across test splits with different S-Levels.
Figure presents the results, with each column representing a different evaluation metric and each row corresponding to a complex system. Within matrix, rows represent evaluation scores for a DNN trained at a specific S-Level, and columns show scores for each test split at corresponding S-Levels. Only the ECE matrix shows clear diagonal behavior, indicating optimal scores when training and test S-Levels match. MSE shows partial diagonal trends, but its Refinement component weakens this pattern (33.22)) while AUC-PR shows no such trends. In $F.3 we confirm the generality of these findings across additional classification metries (Precision, Recall, F1-Score, AUC-ROC), scoring rules (BCE, CRPS, Energy Score), and a spatial correlation- based metric.
0.98
0.96
Train S-Level
20
0.175
0.150
Train S-Level
20
0.150
0.175
Train S-Level
10
20
0.94
0.125
0.125
0.100
0.100
0.90
0.075
0.075
0.88
0.050
0.050
0.86
0.025
0.025
0.84
0.000
0.000
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
Time Step
Time Step
Time Step
Figure 4: Two DNNs were trained on 700 forest fire simulations with different S-Levels-10 (orange, low stochasticity) and 20 (blue, high stochasticity) and evaluated on 300 test simulations with S-Level 20. Evaluation metrics include (a) AUC-PR, (b) MSE, and (c) ECE, measured over an extended prediction horizon. AUC-PR shows similar trends for both models, failing to distinguish the stochastic mismatch. MSE exhibits a steeper decline for the mismatch case but also degrades for both models. In contrast, ECE remains low and stable for the DNN trained on S-Level 20, highlighting its unique ability to track alignment with the Statistic-GT. This behavior also highlights the potential stability in long-horizon predictions when tracking Statistic-GT, as it represents single underlying property of the system, unlike Observed-GT which varies across multiple possible outcomes.
4.2.2 LONG HORIZON BEHAVIIR OF ECE
As discussed in |3.3 evaluating fidelity to the Statistic-GT offers potential long-term stability due to the presence of a single Statistic-GT, unlike the wide range of Observed-GTs. We test whether this stability is reflected in the behavior of the evaluation metric. We hypothesize that ECE will remain low and stable over time when the DNN has learned the correct Statistic-GT, unlike traditional metrics which focus on Observed-GTs, Conversely, if there is mismatch in the Statistic-GT, only ECE can exclusively capture this discrepancy, unlike bascline metrics.
Figure 4 shows two DNNs trained on different S-Levels (10 and 20) and tested on S-Level 20. For both DNNs, baselines like AUC-PR and MSE exhibit similar declining trends as prediction horizon increase, with MSE showing some distinction masked by system variance, and AUC-PR completely failing to differentiate between the models entirely. In contrast, BCE remains low and stable for the DNN trained on S-Level 20, indicating it has learned the correct Statistie-GT. Further exploration in F.4 exhaustively confirms across all S-Levels that while baseline metrics decline with increasing S-Level and prediction horizon, ECE remains stable due to its alignment with the Statistic-GTT
What makes ECE different from the baseline metrics? Uncertain predictions at the micro level can collectively reveal macro-level properties like the Statistic- GT. ECE bins the predictions in I and then compares the average predicted score 6a with the empirical fraction of positives frac(k) (k) in each bin. This property of ECE calculation allows for flexibility in micro-level mismatches while focusing on macro-level calibration, effectively capturing fidelity to the Statistic-GT (see $D.2).
8

---

Published as a conference paper at ICLR 2025
5 CASE STUDY: REAL WORLD WILDFIRE PREDICTION
In this section, we apply our findings to the publicly available Next Day Wildfire Spread (NDWS) dataset, large-scale multivariate dataset of U.S. wildfires aggregated via Google Earth Engine (Huot cet al. 2022) (see G.I for details on wildfire modeling). Unlike our synthetic experiments, where the S Level provided precise control over stochasticity, the NDWS dataset involves unpredictable factors such as weather, vegetation, and human interventions, making it impossible to manipulate or quantify stochasticity. Given this limitation, we reassess prior work the NDWS dataset, analyzing differences in the behavior of evaluation metrics like ECE, AUC-PR, and MSE, and interpret these observations in light of our benchmark experiments. We also explore how ECE can be integrated into the current evaluation protocol to help resolve conflicts in DNN ranking.
firo-mask (1)
Pre-mmask (++1) Observed GT
predicted fire-mask
Evaluation Metrics for Conv-AE stratified by Dice Coefficient values measuring Fire Map Overlap (FMO) between tt and t + 1th day. (t means higher is better, + means lower is better. Colors (normalized column-wise) indicate relative performance, with red rep- resenting worse scores and lighter shades representing better scores. ECE trends differently from F2R metrics.
Table 2:
AUG-tc coeE
6067
FMO (DC)
Sup.
Precision"
Recall
AUC-PR
VSS
ECE
DC: 0.64
AUCUPR 0,526
0.9-10
0.000
0.000
1.000
0.000
0.001
0.8-0.9
0.000
0,000
0.445
0.001
0.006
0.7-0.8
0.771
0,578
0.693
0.003
0.008
Figure 5: DNN takes as input a fire mask
0.6-0.7
0.571
0,700
0.624
0.013
0.019
(left column) and 11 observational vari-
0.5.0.6
116
0.475
0.606
0.512
0.019
0.025
0.400.5
145
0.373
0.501
0.355
0.017
0.022
ables to predict the next-day fire mask (right column), compared to Observed- GT (middle column). F2R metrics indi- cate suboptimal performance.
0.3-0.4
142
0.324
0.381
0.283
0.019
0.022
0.2-0.3
217
0.281
0,284
0.217
0.017
0.014
0.1-0.2
150
0.206
0.121
0.128
0.017
0.013
0.0-0.1
836
0.085
0.028
0.044
0.007
0.005
Overall
1689
0.346
0,311
0.247
0.012
0.012
Forecasting problem. The NDWS dataset compiles historical wildfire incidents into two-dimensional grids at km resolution, with 11 observational variables: elevation. wind direction and speed, temperature extremes, humidity, precipitation, drought index, vegetation, population density, and energy ($G.2) (Huot et al., (022).. The dataset includes 18.545 wildfire events, providing sequential snapshots of fire spread at times and 1++1 1 day. A convolutional autoencoder- based DNN architecture, Conv-AF, is trained and tested on the data. The DNN takes as input a spatial map of the 11 variables along with the fire spread at time & and outputs a binary map predicting fire spread at time +++ ++1. 1.
Observations in evaluation. Huot al. reported an AUC-PR score of 0.284 for Conv-AE on the test split. They noted that "while the metrics the positive class seem low," qualitative visualizations showed that "fires are predicted," " "predicted fires are roughly in the target location." " and there is shown in Figure 5. displaying the fire mask at time t, the next-day fire mask at time t + 1 (Observed -T),, and the Conv-AE forecast. Despite the low scores suggesting poor performance, qualitative insights indicate otherwise. We hypothesize that the highly stochastic nature of wildfires limits the effectiveness of classification-based metrics, prompting reevaluation of the DNN's performance.
"good recognition of larger fires" (Huot et al. 2022). Similar observations
Revisiting evaluation of Conv-AE. Huot et al. reported Precision, Recall, and AUC-PR for their model. We stratify these metrics by the Dice Coefficient (DC) between fire masks at times / and (+ and report the scores in Table Higher DC values indicate gradual fire progression, while lower values signify abrupt changes, often observed in smaller fires. In Figure DC of 0.08 represents a complete fire front shift, while a DC of 0.64 shows slower progression. Table reveals distinct trends across the evaluation metrics. Classification-baat based metries (Precision, Recall, and AUC-PR) degrade significantly as DC decreases, particularly in the lowest DC range (0.0-0.1). I). In contrast, MSE and ECE exhibit different behaviors. While MSE balances aspects of classification metrics and ECE, the latter shows notable improvement in performance as DC decreases. The calibration curve GG.3) confirms that the DNN's predictions are cell-calibr at probability extremes, despite some overconfidence in the mid-range. Hence, ECE behaves differently from classification-bet based metrics as DC varies, and therefore measures a complementary aspect performance. We attribute this behavior to ECE's exclusive focus on testing F2SP, demonstrated in our benchmark experiments.

---

Published as a conference paper at ICLR 2025
Bridging metric conflicts. In G.44 we simulate selecting the optimal DNN from five DNNs trained on the NDWS dataset and observe rank conflicts between AUC-PR, MSE, and ECE. Such conflicts common in model development, with no clear strategy for metric prioritization (Heaton et al.) 2018). To address this, we introduce a cohesive evaluation framework shown in Figure [ The framework places AUC-PR the y-axis, aligning with F2R, and ECE the x-axis, aligning with F2SP, reflecting their complementary roles. MSE sits between them, balancing both criteria. This framework ensures models are first validated for their understanding of stochastic dynamics (F2SP) before assessing accuracy for specific outcomes (F2R). A practical guide is provided in SH
6 RELATED WORKS
Our interdisciplinary study bridges the application of deep learning to complex stochastic systems and their evaluation. While deep learning plays crucial role in forecasting for fields like epidemiology. finance, weather, and geophysics, current evaluation strategies neglect the stochastic nature of these systems, focusing solely on F2R (see CC.3). This limitation extends to computer vision tasks-where many methods used in complex systems originate- suchh as stochastic video prediction (regression) and segmentation map forecasting (classification), where evaluation predominantly relies on the F2R strategy (see $C44. Our work addresses this gap by introducing the evaluation criteria of testing F2SP. We also contribute to sensitivity analysis of evaluation metrics, extending prior studies on simple, low-dimensional setups with standard distributions (e.g., Normal, Exponential) to high- digensinnns environments using stochasticity as the sensitivity variable (see CC.5
7 CONCLUSION, LIMITATIONS, AND FUTURE WORK
We propose a new evaluation criterion to assess a DNN's s ability to capture the stochastic interactions driving the evolution of complex systems. Through controlled experiments in synthetic framework, we show that ECE uniquely evaluates this capability compared to classification-basse based metrics and proper scoring rules. Notably, ECE remains stable in long-horizon prediction performance, unlike other metrics. Applying these insights to a real-world wildfire dataset, we address the disconnect between positive qualitative assessments and negative performance scores and discuss an evaluation framework to mitigate rank conflicts between metrics. While fully understanding complex systems remains challenge, they are crucial due their societal impact. Our work highlights the need for improved DNN evaluation in stochastic scenarios, paving the way for more robust strategies.
Limitations of ECE. While ECE is effective for testing F2SP, it has limitations when compared to F2R evaluation strategy. Specifically, ECE has lower discriminative power than classification-band based metrics in distinguishing between DNNs ($F3). This limitation arises because calibration error lacks refinement term to measure prediction sharpness, critical aspect of F2R evaluation ($3.4.2) Further, ECE requires a sufficient number of samples for reliable convergence. In our experiments, which focused on multivariate problems, a test batch size of 10 grids (each 64 x 64) was sufficient for convergence (fF.)). Future work could explore improved calibration error estimators, such as those in (Gruber & Buettner 2022), which offer stronger guarantees with smaller test sets. This approach could extend our study to uni-variate forecasting tasks, where sample size will be key constraint.
Applications beyond binary classifications and future work. This paper focuses on binary and discrete prediction tasks, leaving extensions to regression tasks for future work. While calibration intuitive for clasifiiation-wheer predicted probabilities should match observed frequencies-it becomes more complex for regression tasks (Kuleshov et al. 2018 Levi et al.. 2022). Future work can explore adapting our findings to regression problems. While this work focuses on complex systems, it would be important to explore if the F2SP evaluation strategy benefits vision problems discussed in CC44 Although such tasks are not traditionally classified as complex systems, their problem formulation and local pixel interactions in videos provide compelling analogy.
key limitation of the NDWS dataset is its restriction to next-day predictions, preventing long-term ECE tracking. Additionally, the absence of open-source complex system datasets- in fields like Computer Vision and Natural Language Processing- hampers broader experimental validation. Future efforts should prioritize collecting and standardizing large-scale datasets for complex systems, with data spanning multiple time steps to enable long-term ECE monitoring. Assessing ECE stability and its potential benefits over time is key avenue for future research.
10

---

Published as a conference paper at ICLR 2025
ACKNOWLEDGMENTS
The authors thank Hemant Kumawat and Minah Lee for their insightful discussions, which signifi- cantly contributed to shaping the ideas presented in this paper. This work was supported in part by the Office of Naval Research under Grant N00014-20-1-225
The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Office of Naval Research or the U.S. Government.
REFERENCES
Carol Alexander, Michael Coulon, Yang Han, and Xiaochun Meng. Evaluating the discrimination ability of proper multi-variate scoring rules. Annals of Operations Research, 334(1):857-883,
2024.
Daniel Arp. Erwin Quiring, Feargus Pendlebury, Alexander Warnecke, Fabio Pierazzi, Christian Wressnegger, Lorenzo Cavallaro, and Konrad Rieck. Dos and don'ts of machine learning in computer security. In 31st USENIX Security Symposium (USENIX Security 22), pp. 3971-3988, 2022.
Per Bak, Chao Tang. and Kurt Wiesenfeld. Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett, 59:381-384, Jul 1987. doi: 10.1 Rtw.e//hhhyyeLLttt...9... URL https
//link. aps wwwg://///ooiooooo 1ttp://1...11/0PPP. 1tt..11111s///yyssetttttte...e.e 381]
M. Belem and M. Saqalli. Development of integrated agent-based model for the management of protected area of tanzania. Land Use Policy, 67:472-486, 2018. doi: 10. 04.055.
Gail Blattenberger and Frank Lad. Separating the brier score into calibration and refinement compo- nents: A graphical exposition. The American Statistician, 39(1):26-33 32, 1985. ISSN 00031305.
URL http: /www. jstor. stable
Rafael Bomfim, Sen Pei, Jeffrey Shaman, Teresa Yamana, Hernán A Makse, Jose Andrade Jr.
Antonio S Lima Neto, and Vasco Furtado. Predicting dengue outbreaks neighbourhood level using human mobility in urban areas. Journal of the Royal Society Interface, 17(171):20000611, 2020.
Massimo Bonavita and Patrick Laloyaux. Machine learning for model error inference and correction. Journal of Advances in Modeling Earth Systems, (2(1))22020MS000222. 2020.
A. Crooks, C. Castle, and M. Batty. Key challenges in agent-based modelling for geo-spatial simulation. Computers, Environment and Urban Systems, 64:13-24, 2017. doi: 10.1016/j.
compenvarbsss.2017....4
John Denker and Yann LeCun. Transforming neural-net output levels to probability distributions. Advances in neural information processing systems, 3, 1990.
Jigar Doshi, Dominic Garcia, Cliff Massey, Pablo Llueca, Nicolas Borensztein, Michael Baird, Matthew Cook, and Devaki Raj. Firenet: Real-time segmentation of fire perimeter from acrial video. In Newral Information Processing Systems 2019, 2019. Available: https::/
htt//bbs/19100 06407
Peter Eastman. Introduction statistical mechanics.
http:://ee.ssaaffordd.
-eeastman/sttet
sttpmech/tttct/ct/o Desastman//tattmechh// 2015. URL
statmech/ Accessed: May 1st, 2024.
J. D. Farmer and D. Foley. The economy needs agent-based modelling. Nature, 460:685-68,, 2009. doi: 10.1038/460685a.
C. Ferri, J. Hernández-Oralle and R. Modroiu. An experimental comparison of performance measures for classification. Partern Recognition Letters, 30(1):27-38, January 2009. ISSN 200. 0167-8655, doi: 1ott 010. URL https:/ www. www.ncedirect.ccoom sci oet.anee//rriiccee///77771115550

---

Published as a conference paper at ICLR 2025
Mark A. Finney. FARSITE: Fire area simulator- development and evaluation. Technical report,
1998. URL httpttt:://b...r///100 2t1.:...2000111...77222221..0 273.22Prmrssss-----. rp--
Damien Fourure, Muhammad Usama Javaid, Nicolas Posocco, and Simon Tihon. Anomaly Detection: How to Artificially Increase Your FI-Score with a Biased Evaluation Protocol. Yuxiao Dong. Nicolas Kourtellis, Barbara Hammer, and Jose A. Lozano (eds.), Machine Learning and Knowledge Discovery in Databases. Applied Data Science Track, Lecture Notes Computer Science, pp. 3-18, Cham, 2021. Springer International Publishing. ISBN 978-3-030-86514 14-6. doi: 10.1007)
978-3-030-86514-6_1
Yarin Gal and Zoubin Ghahramani. Dropout bayesian approximation: Representing model uncertainty deep learning. In international conference on machine learning, pp. 1050-1059. PMLR, 2016.
Robert G. Gallager. Stochastic Processes: Theory for Applications. Cambridge University Press, 2013. doi: 00007//B8097811396665
Zhangyang Gao, Cheng Tan, Lirong Wu, and Stan Li. Simvp: Simpler yet better video prediction. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pp- 3170-3180, 2022.
Tilmann Gneiting and Adrian E Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American statistical Association, 102(477):359-378 2007.
Joana Gouveia Freire and Carlos Castro DaCamara. Using cellular automata to simulate wildfire propagation and to assist fire management. Natural Hazards and Earth System Sciences, 19(1): 169-179, January 2019. doi: 10.5194/nhess-
1etihhess----1--6- 1e----9-99999-99 199-99900..
Gerrit Großmann, Michael Backenköhler, and Verena Wolf. Importance interaction structure and stochasticity for epidemic spreading: A covid-19 case study. In Quantitative Evaluation of Systems: 17th International Conference, QEST 2020, Vienna, Austria, August 31 31-September 2020, Proceedings 17, pp. 211-229. Springer, 2020.
Sebastian Gruber and Florian Buettner. Better uncertainty calibration via proper scores for classifica- tion and beyond. Advances in Neural Information Processing Systems, 35:8618-8632, 2022.
Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In International conference on machine learning, pp. 1321-1330. PMLR, 2017.
William W Hargrove, RH Gardner, MG Turner, WH Romme, and DG Despain. Simulating fire
patterns in heterogeneous landscapes. Ecological modelling, 135(2-3):243-222 2000.
Matthew Heaton, Abhirup Datta, Andrew Finley, Reinhard Furrer, Rajarshi Guhaniyogi, Florian
Gerber, Robert B. Gramacy, Dorit Hammerling, Matthias Katzfuss, Finn Lindgren, Douglas W. Nychka, Furong Sun, and Andrew Zammit-Mangion. A case study competition among methods for analyzing large spatial data, 2018.
Mikael Henaff, Junbo Zhao, and Yann LeCun. Prediction under uncertainty with error-encoding networks, 2017.
Steven A Hicks, Inga Strümke, Vajira Thambawita, Malek Hammou, Michael A Riegler, Pål Halvorsen, and Sravanthi Parasa. On evaluation metrics for medical applications of artificial intelligence. Scientific reports, 12(1):5979, 2022.
N. Hoertel, M. Blachier, C. Blanco, M. Olfson, M. Massetti, M. S. Rico, F. Limosin, and H. Leleu. A stochastic agent-based model of the sars-cov-2 epidemic in france. Nature Medicine, 26:1417-14211 2020. doi: 10. N0.1008/44159 1591-0200000 1001-6.
Fantine Huot, R. Lily Hu, Nita Goyal, Tharun Sankar, Matthias Ihme, and Yi-Fan Chen. Next day wildfire spread: A machine learning dataset to predict wildfire spreading from remote-sensing data. IEEE Transactions on Geoscience and Remote Sensing. 60: 1-13, 2022. doi: 10.1.00/ 3192974.
12

---

Published as a conference paper at ICLR 2025
Mohamed R Ibrahim, James Haworth, Aldo Lipani, Nilufer Aslam, Tao Cheng, and Nicola Christie. Variational-Issie autoencoder to forecast the spread of coronavirus across the globe. PloS one, 16 (1):00246120, 2021.
Piyush Jain, Sean C.P. Coogan, Sriram Ganapathi Subramanian, Mark Crowley, Steve Taylor, and Mike D. Flannigan. A review of machine learning applications in wildfire science and management. URL https:://o1
Environmental Reviews, 28 28(4):478-505 2020. doi: 10.
org/10. 11000..119/ee. 2020- 10002-00111 0019
Xiaoqian Jiang. Melanie Osl, Jihoon Kim, and Lucila Chno-Machado, Calibrating predictive model estimates support personalized medicine. Journal of the American Medical Informatics Association, 19(2):263-21) 2012.
Beomseok Kang, Harshit Kumar, Minah Lee, Biswadeep Chakraborty, and Saibal Mukhopadhyay. Learning locally interacting discrete dynamical systems: Towards data-efficient and scalable prediction. In Proceedings of the 6th Annual Learning for Dynamics & Control Conference, volume 242 of Proceedings of Machine Learning Research, pp. 1357-1369. PMLR, 2024. URL Learning for Dynamics and Control Conference
https: //rx/aaavvv./..///////aaa/aaiai/ org/ abs atg/abs///e (L4DC) 2024.
06460
Ravikiran Keshavamurthy, Samuel Dixon, Karl T Pazdernik, and Lauren E Charles. Predicting infectious disease for biopreparedness and response: A systematic review of machine learning and deep learning approaches. One Health, 15:100439, 2022.
Alireza Koochali, Peter Schichtel, Andreas Dengel, and Sheraz Ahmed. Random noise vs. state-of- the-art probabilistic forecasting methods: case study on crps-sum discrimination ability. Applied Sciences, 12(10):5104 2022.
Volodymyr Kuleshov, Nathan Fenner, and Stefano Ermon. Accurate uncertainties for deep learning using calibrated regression. In International conference on machine learning, pp. PMLR, 2018.
Harshit Kumar, Nikhil Chawla, and Saibal Mukhopadhyay. Towards improving the trustworthiness of hardware based malware detector using online uncertainty estimation. In 2021 58th ACM/IEEE Design Automation Conference (DAC), pp. 961-966. IEEE, 2021.
Cheng-Pin Kuo and Joshua Fu. Evaluating the impact of mobility covid-19 pandemic with machine learning hybrid predictions. Science of The Total Environment, 758: 144151, 2021.
James Ladyman, James Lambert, and Karoline Wiesner. What is a complex system? European Journal for Philosophy of Science, 3:33-67, 2013.
Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in neural information processing systems, 30, 2017.
Dan Levi, Liran Gispan, Niv Giladi, and Ethan Fetaya. Evaluating and calibrating uncertainty prediction in regression tasks. Sensors, 22(15):5540, 2022.
Jiang-Cheng Li, Chen Tao, and Hai-Feng Li. Dynamic forecasting performance and liquidity evaluation of financial market by econophysics and bayesian methods. Physica A: Statistical Mechanics and its Applications, 588: 126546, 2022.
Naian Liu, Jiao Lei, Gao Wei, Haixiang Chen, and Xiaodong Xie. Combustion dynamies of large-scale wildfires. Proceedings of the Combustion Institute, 38, 01 2021. doi: 10. (0.1016/j.proci.22222 1.006.
Edward Lorenz. The butterfly effect. World Scientific Series on Nonlinear Science Series A, 39: 91-94, 2000.
Pauline Luc, Natalia Neverova, Camille Couprie, Jakob Verbeek, and Yann LeCun. Predicting deeper into the future of semantic segmentation. Proceedings of the IEEE international conference on computer vision, pp. 648 657, 2017.
13

---

Published as a conference paper at ICLR 2025
Pauline Luc, Camille Couprie, Yann Lecun, and Jakob Verbeek. Predicting future instance segmenta- tion by forecasting convolutional features. In Proceedings of the european conference on computer vision (ECCV), pp. 584-599, 2018.
Steven M Manson, Shipeng Sun, and Dudley Bonsal. Agent-based modeling and complexity. Agent- based models of geographical systems, pp. 125-139, 2012.
Étienne Marcotte, Valentina Zantedeschi, Alexandre Drouin, and Nicolas Chapados. Regions of reliability in the evaluation of multivariate probabilisticc forecasts. In Proceedings of the 40th International Conference on Machine Learning, ICML'23. JMLR.org, 2023.
Michael Mathieu, Camille Couprie, and Yann LeCun. Deep multi-scale video prediction beyond mean square error, 2016.
Mahdi Pakdaman Naeini, Gregory Cooper, and Milos Hauskrecht. Obtaining well calibrated proba- bilities using bayesian binning. In Proceedings of the AAAI conference on artificial intelligence, volume 29, 2015.
Jerzy Neyman and Egon Sharpe Pearson. Ix. on the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character,
-379, 1933.
Jeremy Nixon, Michael Dusenberry, Linchuan Zhang, Ghassen Jerfel, and Dustin Tran. Mcasuring calibration in deep learning. In CVPR workshops, volume 2, 2019.
Sergiu Oprea, Pablo Martinez-Gonzzlez,, Alberto Garcia-Garcia John Alejandro Castro- Vargas, Castro-Vargas, Sergio Orts-Escolaao, Jose Garcia-Rodriguez, and Antonis Argyros. A review on deep learning techniques for video prediction. IEEE Transactions on Pattern Analysis and Machine Intelligence, 4466))2806-2826, 2022. doi: 10.1 1A.11/9//PAM1222030455500
Pierre Pinson and Julija Tastu. Discrimination ability of the energy score. 2013.
David Radke, Anna Hessler, and Dan Ellsworth. Firecast: Leveraging deep learning to predict wildfire spread. In Proceedings of the 28th International Joint Conference on Artificial Intelligence, IJCAI' 19, pp. 3575-4581. AAAI Press, 2019. ISBN 9780999241141.
Daniel Ramos, Javier Tranco-Pedross Alicia Lozano-Diet: and Joaquin Gonzalez-Rodriguer De- constructing cross-entropy for probabilistic binary classifiers. Entropy, 20(3):208, 2018.
Rebecca Roelofs, Nicholas Cain, Jonathon Shlens, and Michael C Mozer. Mitigating bias in calibration error estimation. In International Conference on Artificial Intelligence and Statistics, pp. 4036-4054. PMLR, 2022.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation, 2015.
Richard C Rothermel. A mathematical model predicting fire spread in wildland fuels, volume 115. Intermountain Forest & Range Experiment Station, Forest Service, ...... 1972.
Hiroki Sayama. Pyex: python-based simulation code repository for complex systems education. Complex Adaptive Systems Modeling, 1:1-10, 2013.
Sebastian Scher and Gabriele Messori. Ensemble methods for neural network-based weather forecasts.
Journal of Advances in Modeling Earth Systems, 13(2), 2021.
J. Schulze, B. Müller, J. Groeneveld, and V. Grimm. Agent-based modelling of social-ccological systems: Achievements, challenges, and way forward. Journal of Artificial Societies and Social Simulation, 20(2):8, 2017. doi: 10. 10. 00.18564/jass.3422)
Murat Sensoy, Lance Kaplan, and Melih Kandemir. Evidential deep learning to quantify classification uncertainty, Advances in neural information processing systems, 31, 2018.
Xingjian Shi, Zhourong Chen, Hao Wang, Dit- Yan Yeung, Wai kin Wong, and Wang chun Woo. Convolutional Istm network: A machine learning approach for precipitation nowcasting, 2015.
14

---

Published as a conference paper at ICLR 2025
Helen R. Sofaer, Jennifer A. Hoeting, and Catherine S. Jarnevich. The area under the precision-reeall recall curve as a performance metric for rare binary events. Methods in Ecology and Evolution, 10(4))565- 577, 2019a. doi: http://wwww.aalcccoooo/o//aaaaaa
URL htt.h:tss//// /ttpttt:::///ss/oooee jerrnnaasoo
on in wwweeiiibaar ibbrrr. wi ey. com doi atpoi/abs/10..o
21 10. 13140)
Helen R Sofaer, Jennifer A Hoeting, and Catherine S Jarnevich. The area under the precision-recall curve as a performance metric for rare binary events. Methods in Ecology and Evolution, 10(4): 565-577, 2019b.
O.
and J. Margerit. Modelling forest fires. part i: A complete set of equations derived by extended irreversible thermodynamics. International Journal of Heat and Mass Transfer, 45: 1722, 04 2002. doi: 10.1016/50017-991
Adrian Tasistro-Hart, Alexander Grayver, and Alexey Kuvshinov. Probabilistic geomagnetic storm forecasting via deep learning. Journal of Geophysical Research: Space Physics, 126 (1)=e2020AA028228, 2021.
Matthew Thompson, Phil Bowden, April Brough, Joe Scott, Julie Gilbertson-Day, Alan Taylor, Jennifer Anderson, and Jessica Haas. Application of wildfire risk assessment results to wildfire response planning in the southern sierra nevada, califormia, usa. Forests, 7(12):44, Mar 2016. ISSN 1999-4907. doi: 10.3390/7030064, URL wttp::t///// //tp:://xxxx// dww.dxxoooo...ooooo org/ 10. 3. 3390/
Seth Tisue and Uri Wilensky. Netlogo: A simple environment for modeling complexity. In Interna- tional conference on complex systems, volume 21, pp. 16-21. Citeseer, 2004.
Yunbo Wang, Mingsheng Long, Jianmin Wang, Zhifeng Gao, and Philip S Yu. Predrnn: Recurrent neural networks for predictive learning using spatiotemporal Istms. Advances in neural information processing systems, 30, 2017.
Yunbo Wang, Zhifeng Gao, Mingsheng Long, Jianmin Wang, and S Yu Philip. Predrnn++: Towards a resolution of the deep-in- in-time dilemma in spatiotemporal predictive learning. In International Conference Machine Learning, pp. 5123-5132. PMLR, 2018
Yi-ming Wei, Shang-jun Ying, Ying Fan, and Bing-Hong Wang. The cellular automaton model of investment behavior in the stock market. Physica A: Statistical Mechanics and its Applications, 325(3-4):507-516, 2003.
A. L. Westerling, H. G. Hidalgo, D. R. Cayan, and T. W. Swetnam. Warming and ear- lier spring increase western u.s. forest wildfire activity. Science, 31 313(5789);940 943, 2006. doi: 10..112//ccence..11200 1228844. TRRLhttps::// www.:sccencee. watpeeee///d////aa/////aa/// science. 1128834
Enze Xie, Wenhai Wang, Zhiding Yu, Anima Anandkumar, Jose M. Alvarez, and Ping Luo. Segformer: Simple and efficient design for semantic segmentation with transformers, 2021.
Reda Yacouby and Dustin Axman. Probabilistic extension of precision, recall, and fll score for more thorough evaluation of classification models. In Proceedings of the first workshop on evaluation and comparison of NLP systems, pp. 79-91, 2020.
Suwei Yang, Massimo Lupascu, and Kuldeep Meel. Predicting forest fire using remote sensing data and machine learning. Proceedings of the AAAI Conference on Artificial Intelligence, 03 2021. doi: 10. v35.m..11777
Siwei Yu and Jianwei Ma. Deep learning for geophysies: Current and future trends. Reviews of Geophysics, 2003) 2021.
Richard Zinck and Volker Grimm. More realistic than anticipated: A classical forest-fire model from statistical physics captures real fire shapes. The Open Ecology Journal, 1:8-13, 09 2008. doi: 10.2174/1 07114//08871111000000000000000
15

---

Published as a conference paper at ICLR 2025
A
OVERVIEW OF THE SUPPLEMENTARY MATERIAL
Notations used in the paper
Literature Survey
Application of Agent-based Models (ABMs) in complex systems.
General use of ECE as an evaluation Metric vs. our work
Complex systems, their Prevalence, and evaluation metrics
Evaluation strategy for multivariate forecasting in computer vision
Sensitivity study of evaluation metrics
.
Background
Description of the synthetic complex environments
Proper scoring rules penalize uncertain predictions at micro scale
D.2
DNN Characterization
Design rationale behind convLSTM- M-C and conv-CA
Impact of changing S-Level on Statistic-GT and DNN predictions
Additional Results
Impact macro-varianee on Fidelity Realization strategies
Calibration Curve of the perfect predictor predicting Statistic-GT
Generalizability of ECE's diagonal behavior
Long horizon behavior of ECE vs. baselines
ECE convergence vs. Number of samples
Application of Deep Learning in Wildfire Modeling
Models for wildfire modeling.
Observational variables in Next Day Wildfire Spread (NDWS) dataset
Calibration Curve NDWS dataset
Resolving rank conflicts in DNN rankings
A practical user guide for evaluating complex systems
B NOTATIONS USED IN THE PAPER
Table 3: Summary of notations used in this study
Symbol
Description
W
Grid size
Location of a single cell within the grid
State of a cell at time
A specific state of interest
present
absent
time in cell (f,j)
B,
Observed GT: System state across all cells time f, B, (hr,a..))) Jow
Oz
n-dimennioonll vector of observational variables across grid at time 1, O. (Roуlio Predicted probability of state presence for cell (f.j) at time
P.
Predicted joint conditional distribution of system states
PL(5))
Probahiiity of state actually present cell (1. at time
Mc Bem(P.(e))
Micro random variable modeling presence/absence cell (i. at time with parameter Pi Statistic- GT: Joint probability distribution of Macro random variable representing the collective state the system at time f Evaluation metric
all cells
2,
S(.-)
16

---

Published as a conference paper at ICLR 2025
C LITERATURE SURVEY AND RELATED WORKS
C.I APPLICATION OF AGENT- BASED MODELS (ABMS) IN COMPLEX SYSTEMS.
ABMs, with each pixel acting as an individual agent, offer natural way to simulate stochastic inter- actions. These models, defined by discrete space and time, and marked by local spatial interactions, align well with the physical nature of complex systems, e.g., (Gouveia Freire & Castro DaCamara 2019). For example, in ABM forest fire models, each cell on fire is an agent capable of spreading the fire based on neighborhood interaction rules, leading to emergeni behaviors that mirror real-tife fire propagation patterns (Zinck& & Grimm 2008 Bak et al. 1987). In general, ABM has been applied to wide range of complex systems in recent years. Key application areas include social-ecologicat systems, where ABMs have been used to study land-use changes, environmental adaptation, and biodiversity protection (Schulze et al. (2017). In urban planning and traffic simulation, ABMs have modeled city growth, sustainability, and traffic flow dynamics Crooks et al. (2017). The COVID-19 pandemic has driven increased use of ABMs in public health and epidemiology for simulating disease spread and evaluating intervention strategies Hoertel et al. (002)). In economics and finance, ABMs have been employed to model market dynamics and assess economic policies /arieer//Ferrrra & Foley| (2009). Environmental management has benefited from ABM applications in forest management and climate change adaptation studies Belemm& & Saqalli (208)). These diverse applications demonstrate the versatility and power of ABM in capturing the intricacies of real-world complex systems.
C.2 GENERAL USE OF ECE AS AN EVALUATION METRIC VS. OUR WORK
ECE is widely used to evaluate the calibration of DNNs in terms of confidence estimates and uncertainty quantification. Typically applied in static tasks such image or text classification and object detection, ECE focuses on univariate predictions, where each test sample corresponds to a single outcome and prediction. In these contexts, studies have shown that DNN's confidence is influenced by training and architectural factors. For instance, (Guo et al. 2017) observed that model capacity, batch normalization, and regularization techniques often lead to miscalibrated confidence estimates. Another significant challenge arises with out-of- of-distribution samples, where a DNN's s lack of knowledge impacts the reliability of its confidence scores, prompting research into uncertainty quantification to address this limitation Gaa & Ghahramani 2016).
Techniques such as temperature scaling (Guo et al., (017), ensemble methods (Takshminarayyaan et al.] 2017). Bayesian neural networks (Denker & LeCun 1990), dropout (Gal & Ghahramani| 2016), and evidential deep learning (Sensoy et al. 2018)) to name few, aim to enhance calibration, with ECE serving as the standard metric for assessing calibration quality (Nacini et al. 2015). Additionally, ECE is widely employed in safety-critical domains like healthcare (Jiang et al. 2012) and cybersecurity (Kumar et al.) (202T), where reliable confidence estimates are critical. Importantly, all these prior works focus on addressing the stochasticity in the model's predictive distribution ploutpul (inpul). This stochasticity is model-specific and reflects the uncertainty in DNN's predictions for given input.
However, in complex systems, stochasticity arises from the system's s dynamics (e.g., agent interactions or environmental factors) which fundamentally different from the model's uncertainty about its predictions. ECE's conventional role does not naturally extend to capturing system properties like the Statistic-GT. Beyond measuring calibration error for DNN's s output distribution, we demonstrate that ECE can evaluate F2SP by assessing predictions against the system property Statistic-GT using only Observed-GT7 Our findings establish perfect calibration as a necessary condition for tracking fidelity to the Statistic-GT, a fundamental insight that significantly extends the utility of ECF.
Building on this expanded utility, our work the first to discover ECE's unique utility in evaluating stochastic complex systems with high-dimensional, multivariate outputs, such as the 64x66 grids in our experiments. By addressing the system's ' inherent randomness through F2SP, rather than solely aligning with the Observed-GT as traditional F2R-focused strategies do, our findings position ECE as a complementary metric. While current model development primarily emphasizes discriminative performance through F2R strategy, calibration often takes backseat (Naeini et al., 2015). For forecasting stochastic complex systems, however, calibration must take center stage, as it tests fidelity to fundamental system properties that inherently align with the stochastic nature of these systems.
17

---

Published as a conference paper at ICLR 2025
C.3 COMPLEX SYSTEMS, THEIR PREVALENCE, AND EVALUATION METRICS
Deep learning (DL) is increasingly applied to complex systems, leveraging multi-dimensional data from sources like satellite imagery, population records, and the internet. These datasets enable DL models to uncover intricate patterns crucial for accurate forecasting. Recent applications include infectious disease prediction, where models integrate epidemiologic, geographic, and climatic factors to forecast outbreaks (Keshavamurthy et al., 2022). Ibrahim et al. used LSTM models for COVID-19 forecasts (lbrahim et al. (021), while Bomfim et al. applied mobility data to improve dengue transmission predictions (Bomfim et al. 2020p. Kuo and Fu used county-level data to model COVID- 19 infection rates (Kuo & Fuj 2021). In finance, Li et al. leveraged DL for asset price forecasting, capturing nonlinear dynamics (Li et al., 2022). In weather prediction, scoring rules like MSE and CRPS are commonly used (Scher & Messori 2021 Bonavita & Laloyaux| 2020). DL has also been applied in geophysics. where models predict system states from spoosppaial data (Yu & Ma (0211). Hart et al. used DL to forecast geomagnetic storms using satellite data, evaluating with RMSE, Pearson correlation, and calibration error to assess probabilistic accuracy (Tasistro- Hart et al. 2021).
While most evaluations use probabilistic metrics (e.g. MSE, RMSE, MAE), classification metrics are applied in domains with well-defined outcomes, such as epidemic spread (Bomfim et al., 2020) and wildfire prediction (Huot et al. 2022). However, all approaches treat these problems deterministically through F2R formulation, and do not account for the inherent stochasticity of such complex systems.
C.4 EVALUATION STRATEGY FOR MULTIVARIATE FORECASTING IN COMPUTER VISION
In the computer vision community, evaluation strategies for forecasting tasks across both regression and classification problems predominantly focus on matching the Observed-GT (F2R). For instance, in stochastic video prediction-a regression task-DNNs generate range of potential outcomes, and the score of the sample best matching the Observed-Gi GT is reported (Oprea et al., 20222] Mathieu let al. 2016 Henaff et al. (017). Similarly, for classification tasks such predicting the evolution of segmentation maps, metrics like Intersection over Union and Average Precision (summarizing the
precision-recall curve) are standard (Luc et al. 2017 2018).
While this F2R-centric approach is sufficient for scenarios where stochasticity is sporadic (e.g.. unexpected car turns in video prediction (Oprea et al. 2022 ). it falls short for complex systems where stochasticity is inherent and consistent (Gallager 2013). In such cases, our study emphasizes the importance of matching the Statistic-GT (F2SP) as a more robust evaluation strategy, particularly for high-risk, safety- crriticl applications. By providing stochastic interpretation and stochasticity- compatible evaluation methods, we aim to enhance the reliability of DNNs in such scenarios.
C.5 SENSITIVITY STUDY OF EVALUATION METRICS
Our work aligns with broader research efforts on sensitivity analysis of evaluation metrics across different use cases, highlighting their strengths and weaknesses. MMrcotte et al. (2023) explored how problem dimensionality and Monte Carlo approximation quality affect scoring rules' discriminative capability using synthetic univariate and multivariate distributions (c.g.. Normal, Exponential).
Pinson & Tastu (2013) and Alexander et al. (2024) compared Energy Score with other scoring rules under varying multivariate probabilistic forecast dependencies, modeled with Gaussian Distributions. Koochali al. (2022) identified CRPS-Sum's limitations in distinguishing forecast quality in real- world time series with up to eight variables. Ferri et al. (2009) compared 18 classification-base metrics, analyzing their sensitivity class threshold choice, ranking quality, calibration, and class distribution. Unlike these studies focusing on simple variables, we focus on characterizing evaluation metrics in nigh-dimensional stochastic complex systems, using stochasticity in interactions as the key sensitivity variable. Additionally, a different set of works examines the sensitivity of calibration error to various design choices, such as the binning process (Nixon et al. 2019: Roelofs et al. (022). Unlike these studies, where the sensitivity variable is a property or design choice of the evaluation metric itself, our sensitivity variable is a parameter of the underlying complex system being evaluated.
Further, sensitivity of cesiificcatio-baseed metrics have been studied in other contexts. Metrics like Fl score, precision, and recall are sensitive to evaluation protocols. In anomaly detection and security. factors like train-test splits, base rates, and decision thresholds impact these metrics (Fourure et al. 2021] Arr et al. 2022). In medical applications, Hicks et al. (2022) showed that interpretations of these metrics depend on class distribution and testing methods. In NL.. Yacouby & Axman (020
18

---

Published as a conference paper at ICLR 2025
noted that these metrics evaluate only the top prediction, proposing alternatives. For geo-spatial grids similar to ours, Sofaer et al. (2019b) found AUC-PR more reliable than AUC-ROC due to its insensitivity to dataset imbalance. Contributing to this literature, highlight new key limitation: classification metrics become unreliable in highly stochastic systems.
D BACKGROUND
D.1 DESCRIPTION THE SYNTHETIC COMPLEX ENVIRONMENTS
heattyy
inacivve
infected
dead
emppt
hold
(6)
Figure 6: Figure shows the three different synthetic environments used in this work: (a) Forest Fire Model, (b) Host Pathogen Model, and (c) Stock Market Model. Each system includes four discrete states driven by different temporal dependencies.
(2) fuel ignition
Agent Lifecycle
patch
fire
(1) heat accumulation
radiation कu
cedr
dii + (N))
qfjat
qaie
dead
ember
di < 4in
(4) fuel depletion
Figure 7: Snapshot of forest-fire evolution in NetLoggo, using 64x64 grid of locally interacting agents. Simulation is initialized by randomly assigning each agent in the grid patch (green) or no-patch (black) value and different fire seed locations (3 in our case). Each patch agent goes through four stages, the flowchart of which is depicted in the right. The interaction between the agents leads to the emergent behavior of forest fire evolution. The agent's deterministic evolution rules are inspired from Rothermel's work on heat transfer based model for wildfire evolution (Rothermel) 1972....
Forest Fire Model. Each grid cell (i,j) represents an agent in one of four states St,(1,)
(empty, patch, fire, ember) (denoted 81, 82, 83, 84). The target state consists of the burning states (fire, ember). The simulation starts on 64 64 grid, with each pixel initialized as a 'tree' or 'no-tree'. Agents have heat values ((v,) crucial for the heat transfer in forest fire evolution. Fire seeds, placed at randomized (or fixed) locations, provide initial heat to agents. The initial condition is set ((.r) Leed qureshold for seed locations (i, j). and {(.j) = 0 otherwise, where Qirreshold is the ignition threshold and Iseed amplifies seed heat values. A 'tree' agent accumulates heat from activated neighbors in its Moore neighborhood, in line with heat transfer mechanisms described by (Rothermel 1972), following the equation:
(t + At) (t)
(t). (R.)EN
Σ
The indicator function ((x.)(t)) ensures only 'fire" state agents contribute to heat transfer. An agent's 's heat value ((t,)) exceeding Qhreshood triggers a state change from 'patch' to 'fire", and then "ember" in
19

---

Published as a conference paper at ICLR 2025
the next time step. 'Ember' agents radiate heat rate of qdie to adjacent non-fire patches, gradually losing heat until radiation ceases. This process ends when 'ember' agents darken, indicating ((i.j) falling below certain threshold, thus terminating heat radiation and transitioning to the 'dead' state. The process is summarized in Figure 7
To control the degree of stochasticity in the model, introduce Pignit: the probability of ignition after conditions are met (i.e., q > @threshold). In deterministic scenarios, Pignite 100%, ensuring ignition once the threshold is reached. We define the S-Level parameter (100% Pignite;; ); higher S-Level values indicate more randomness in agent interactions, with S-Level 0 corresponding to deterministic interactions. Figure 2 shows four different Monte Carlo simulations of forest fire evolution.
MC-1
MC-2
MC-3
MC-4
time
Figure 8: Figure shows four distinct Monte Carlo simulations of the evolution in the host pathogen system from the same initial condition.
MC-1
MC-2
MC-3
MC-4
time
Figure 9: Figure shows four distinct Monte Carlo simulations of the evolution in the stock market system from the same initial condition.
Most-Pathogen Model. The Host-Pathogen model (Sayama 2013) simulates virus-host interactions the population level. Each grid cell (i.j) can be one of four states: empty, dead, healthy, or infected. We focus on the healthy state s* = healthy and define Mr,(i.) = [(st,(i.j) = s*). The system evolves as follows: 1) Create a 64 x grid initialized with randomly distributed infected (1%), healthy (75%), and empty (24%) cells. 2) Each dead cell is cured by neighboring healthy cells with probability Peare 0. 15. 3) Each healthy cell is infected by neighboring infected cells with probability Pinfect 0.85. 4) Infected cells transition to dead cells in the next timestep. 5) Repeat steps 2 to 4. This cycle models pathogen spread and recovery dynamics based on probabilistic transition rules. To control the appearance of Statistic-GT,
used Peure the S-Level parameter, varying it between 10%, 15%, and 20%. Figure g shows four different Monte Carlo simulations of system evolution from the same initial condition.
20

---

Published as a conference paper at ICLR 2025
Stock Market Model. The Stock Market model (Wei et al. 2003) uses cellular automata simulate investor behavior influenced by neighboring investors. Each cell (i.j) represents an investor in one of four states: hold, sell, buy, or inactive. We focus the buying state s* buy. defining M,,(t.)) = ((.)) = 8)). The system evolves as follows: 1) Create a 64 x 64 grid with randomly assigned states (buy, sell, hold, inactive). 2) Each cell transitions stochastically based the dominant state of neighboring cells, following a transition matrix parameterized by market status M 0.05 (positive market sentiment) and investment probability Piover 0.95 (Wei et 2003). 3) Cells that have been in the buying state for two consecutive timesteps become inactive. 4) Inactive cells from the previous sequence transition back to buying. 5) Repeat steps 2 to To control the appearance of Statistic-GT. we introduced the S-Level parameter as 100 Pinvest, resulting in S-Levels of 5%, 10%, and 15%. This models investor dynamics influenced by market conditions and peer interactions. Figure shows four different Monte Carlo simulations of system evolution.
D.2 PROPER SCORING RULES PENALIZE UNCERTAIN PREDICTIONS AT MICRO SCALE
Impact Predicted Probability Negative Log Likelihood
NLL for
Negative Log Likeliood (NLL)
Uncertain Predictions are
3-
penalized
2-
Only the high confidence predictions receive zero penalty
00
02
0.4
0.6
0.8
1.0
Predicted Probability (p)
Figure 10: NLL for positive outcomes as function of predicted probability in binary classification problem. NLL penalizes uncertain predictions at the micro-level detail.
To clarify why proper scoring rules cannot test fidelity to the Statistic-GT, consider evaluating single cell within the 64 x 64 grid using Binary Cross-Entropy (BCE), which computes the Negative Log Likelihood (NLL) of the true label y E (0, 1} given the predicted probability p. NLL an optimal metric for evaluating probabilistic forecasts (Neyman & Pearson 1933). calculated as:
BCE(y. ly log(p) + (1 - v) log( - -)].
As shown in Figure 10 NLL penalizes uncertain predictions those with predicted probabilities away from 0 or 1-even if they are accurate in expectation. Only high-confidence predictions receive minimal penalty. Since proper scoring rules promote sharpness, they penalize uncertain predictions at the micro-scale when they do not match the Observed-GT.
In contrast, the bimning process in BCE allows flexibility in micro-scale mismatches as long as the overall predictions are calibrated. This enables ECE to test fidelity to macro properties of the stochastic complex system, such as the Statistic-GT.
E DNN CHARACTERIZATION
E DESIGN RATIONALE BEIIND CONYLSTM-CA AND CONV-CA
We choose ConvLSTM for its ability to model spatiotemporal systems, aligning well with the characteristics of the complex systems. Our modified version (see Figure I1 (a)), COmVLSTM-CA, places a ConvLSTM cell with 3 x 3 kernel between an Encoder (with 3 x 3 kernel) and a Decoder (with 1 kernel). The Encoder takes an RGB image and transforms it into a latent tensor during a 10-timestep observation window. This tensor is then processed by the ConvLSTM cell, maintaining its spatial dimensions, before the Decoder produces burnt map grid of softmax probabilities.
21

---

Published as a conference paper at ICLR 2025
Obsersitc
Pradiciion
Figure 11: (a) Auto-regressive training of the DNN, (b) Softmax probability map from convLSTM with bottleneck, and (c) Softmax probability map from ConvLSTM-CA with no bottleneck.
This design choice preserving spatial dimensions is crucial for modeling the cellular automata application. Reducing the spatial dimensions of the latent tensor is popular design choice in video prediction models such as recurrent neural network-based (Wang al.) 2017 2018) and simple CNN-based models (Gao et al. 2022). This reduction is often employed to decrease computational complexity and to capture essential spatial features while discarding less informative details. However, we observe that adding bottleneck causes individual pixels (agents) to lose their identity, Maintaining these dimensions helps preserve each agent's s identity, a critical factor for developing a DNN that minimizes mis-calibrated forecasts arising from limited model capacity. For instance, seen in Figure 11 (b), using a compressed latent dimension results a forecast cloud around predictions for deterministic fire evolution scenarios, which does accurately reflect the system's true evolution. In contrast, as shown in Figure (c) an uncompressed latent space yields predictions without a forecast cloud, aligning closely with the deterministic system's 's true evolutionary rules. Conv-CA used in is a version of Conv-AE (Huot et al. 2022) with the spatial bottleneck eliminated.
Section G.4
E.2
IMPACT OF CHANGING S S-LEVEL ON STATISTIC-GE GT AND DNN PREDICTIONS
S-Level
S-level: 0
6.7
0.8
Frequency
S-level: 10
0.2
S-level: 20
0.3
000
DNN Forecast
Statistic-GT
Observed-GT
Prediction
(a)
(b)
Figure 12: (a) Visualizations of forest-fire snapshots for different S-Levels (one per row), showing Statistie-GT, Observed-GT, and corresponding DNN predictions (raw forecast values). DNN predic- tions closely resemble Statistic-Gii (b) Histograms showing the frequency of DNN forecast values for different S-Levels. The DNN's predictions become less confident as S-Level increases.
In this section, analyze the impact of changing S-Level Statistic-GT and DNN's 's raw predictions, and further demonstrate that the DNN learns to predict the Statistic-GT in highly stochastic scenarios. For given initial condition (agent configuration, seed location), the S-Level uniquely determines the macro-scale pattern observed in the Statistic-GT. This is evident in the first column of Figure a2 where all three S-Level rows share the same initial condition but differ in S-Level. As S-Level increases, the cloudiness in the macro pattern also increases, confirming that S-Level uniquely determines the macro pattern in the Statistic-GT for given initial condition.
In the second and third columns of Figure 12 and the corresponding DNN prediction on that sample. Higher S-Levels correlate with greater cloudiness in the DNN output. While the DNN's predictions begin to resemble the Statistic-GTT the corresponding Observed-GT can vary significantly and may not match the DNN's prediction.
present the Observed-GT for given MC sample
22

---

Published as a conference paper at ICLR 2025
To further investigate the alignment between the DNN's predictions and the Statistic-Gii we plot the mean Statistic-GT against the corresponding mean DNN forecast in Figure T3 The strong correlation suggests that the DNN is learning to predict the Statistic-GT (see qualitatives in Figure L4p. This capability linked to the use of Binary Cross-Entropy (BCE) loss proper scoring rule which promotes calibrated fore- casts (Gneiting & Raftery, 2007).
1.0
S-Level
Mean Statistie-Gi
0.8
a.a
*-* 10
0.2
0.0
0.0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8 0.9 1.0
As the DNN learns the Statistie-GT,, it begins making more uncertain predictions. Figure 12]b illustrates histograms of DNN predictions across 1000 simulations for each S-Level. In determin- istic settings, predictions are primarily binary (0 or 1), but at S-Level 20, predictions span the entire 0-1 range, indicating increased predictive uncertainty. However, these uncertain predictions still convey valuable information and contribute to larger pattern that emerges the macro scale.
Figure 13: Correlation between Statistic-GT and DNN forecasts, indicating that the DNN is predict- ing the system property Statistic-GTi.
Due to the stochastic evolution, is inherently unachievable for the DNN to replicate the Observed- GT; instead, learns predict the Statistic-GT. If DNN fails predict the Statistic-Gri it indicates that it has learned different stochastic process, making it fundamentally unsuitable for that scenario.
Statistic GT
Observed GT
DNN Forecast
Observed GT
DNN Forecast
Observed GT
Observed GT
DNN Forecast
Figure 14: Qualitative visualizations of the DNN's forecasts across four different Monte Carlo simulations for the forest fire dataset. Frames are shown every five time steps. The DNN observes the first 10 time steps (first 2 frames) and predicts the next 50 time steps (last 10 frames). We can observe the difference between the DNN's forecasts and the Statistic-GTT, which arises because the determinism that is injected into the DNN's predictions due to its observation of the first 10 frames of fire evolution. However, the DNN's predictions closely resemble the evolution of Statistic-GT.
23

---

Published as a conference paper at ICLR 2025
F ADDITIONAL RESULTS
F.1 IMPACT OF MACRO-VARIANCE ON FIDELITY TO REALIZATION STRATEGIES
In this section, show that classification-based metrics like Precision, Recall, and AUC-PR, which test F2R, are highly sensitive to macro variance, exposing the limitations of Observed-Gi as an evaluation target. The section motivates the of Statistic-GT as more stable alternative for assessing model performance in stochastic systems.
Macro Random Variable. To formalize the system-level modeling of stochastic fire evolution, leverage insights from statistical mechanics (Eastman, 2015). At the microscopic level, individual agent behavior is captured through the Micro RV, Mi,(i.)) At the macroscopic level, the system behavior is represented by the Macro RV, Z, Formally, 2, is defined as:
2,
M,,
i=1
Applying the Central Limit Theorem for a large number of Micro RVs (= 10²), Ze can be modeled using a Normal distribution, characterized by its mean E[Z] and variance Var[Zr]. Sampling from Ze provides a macrostate value, representing the aggregate state of all agents. It should be noted that multiple microstates can correspond the same macrostate value. Overall. Var[Zz captures the system's 's tendency to explore diverse macrostates, with higher variance indicating greater overall unpredictability. The parameters of the Macro RV are extracted from the ESP by recording the number of unburnt trees (macrostate) at each time step. Using 1000 MC simulations, we generate distribution at each time step to compute E[Z,] and Var[Z.].
F.1.1
CHARACTERIZING THE S-LEVEL TEST CASES.
Figure 15
time, with colors from red (S-Level=0) black (S- Level=50 50) increments of The central line and shaded areas represent Z[Z and Var Zz].
(a) shows the Macro RV Z,
Figure 15 (b) highlights Var (or[2 [Z] over time for select S-Levels (0, 5, 10. 15, 20) used in our bench-
marks. Key observations: At S-Level 0 (deterministic), the absence of V Varr2,] reflects singular
evolutionary path. Generally, V Varle Var[Z,] increases with S-Level and time, peaking at S-Level 20, indicating maximum variance and chaotic behavior typical of real-world systems (Zinck & Grimm 2008). Beyond S-Level 20, fire dies out, reducing active pixels and Var [Z.]. This setup allows us to test evaluation metrics under varying stochasticity, with S-Level 0 being deterministic and S-L.vel 20 representing peak randomness.
F.1.2 SENSITIVITY OF F2R EVALUATION STRATEGY TO THE SYSTEM MACRO- VARIANCE
Methodology. For this experiment, perform inference the DNN trained with S-Level 20. The test dataset comprises 1000 MC simulations of the S-Level 20 ESP forest fire test case, all
600
S-Level
2500
Hunburnt-ttres
S-Lvvl
400
10
50
Var
1500
0
1000
200
500
100
0000000000000000000000000
25
50
75
1000
125
150
175
200
20
time
time
Figure 15: Characterizes the Macro RV Z, (left) ESP representing Z, across different S-Levels: mean is the center line (Statistic-GT) and variance is the shaded region; (right) Var[Z]] over time for selected S-Level test cases used in this study. S-Level 20 shows chaos (peak variance).
24

---

Published as a conference paper at ICLR 2025
Metric
Precision
Recall
Standard Deviation of Metric
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
50
100
150
200
250
300
350
400
Var(Z,)
Figure 16: SD evaluation metries versus Var[Z,] S-Level 20. Higher SD indicates increased sensitivity to V Var[Zz) Classification-bass metrics (Precision, Recall, AUC-PR) show greater sensitivity in highly stochastic environments, highlighting their unreliability in these settings. In contrast, MSE and ECE exhibit reduced sensitivity as they do not exclusively test F2R.
with identical initial conditions. We calculate the evaluation metric independently for each timestep of each MC simulation. Each simulation given timestep has specific V Yarll Var[Zz) Ze). reflecting macro-variance at that moment. We measure the Standard Deviation (SD) of the evaluation metric across the 1000 MC simulations against V [r[ to assess the metric's sensitivity Since all simulations originate from the same stochastic process, an ideal metric that is faithful to the stochastic process, should demonstrate minimal sensitivity to individual realizations of the ESP.
macro- vocroovariance.
Results. Figure 16 shows the SD of evaluation metrics against V ar Zt]. The significant increase in SD for classinication-bast based metrics (Precision, Recall, AUC-PR) highlights their heightened sensitivity to macro-varte varcinee and reduced reliability. Precision and Recall suffer high variance in stochastic settings due their reliance on thresholding, which can produce predictive outcomes misaligned with the Observed-Gri This issue stems from the incompatibility of "thresholding" with stochastic systems, extending beyond the common critique of arbitrary threshold selection (Fourure) et al. 2021). Although AUC-PR is less sensitive than other wreshold-basel metrics, it encounters challenges from its integration False Positives (FP) through Precision, where the traditional concept of FP loses relevance because any non-zero forecast by the DNN renders both outcomes plausible.
In contrast, MSE and ECE demonstrate lower sensitivity to macro-variance as neither exclusively focuses testing F2R. While both metrics exhibit similar trends in SD vs. Var they converge to different steady-state values. Specifically, the steady-state value of MSE is influenced by Var [t), making function of this variance, whereas ECE's steady-state value remains unaffected by V ar [2] in its convergence (see Figure] 17). This theoretical distinction is further explored in $3.422
25

---

Published as a conference paper at ICLR 2025
0.08-
-
10
15 +
20
0.07
-a-
10
15
20
0.10
0.03
0.04
0.02
0.02
0.01
0.00
150
200
Samples
(a)
(b)
Figure 17: Testing the asymptotic convergence of MSE and ECE. The DNN is trained and tested on the same S-Level. Each sample is grid of 64 64 predictions 1=59 in the test split of the synthetic forest dataset. As the number of samples increase in the calculation of MSE and ECE, they both converge. The MSE converges a value that incorporates the V Var(Z)). while the ECE converges a low value as it excludes the impact of V Var(Z,).
F.2 CALIBRATION CURVE OF THE PERFECT PREDICTOR PREDICTING STATISTIC-GT
S-Level [CCE)
1.0
S-lerel (ECE)
1.0
5-Lerel (CCE)
3.0
S-Lavel [ECEI
S-level [ECE)
0.8-
0:8-
0.8
0.6
0,8-
0.8
0.8
0.0 0.2
0.6.
06
0.8
0.0
0.2
0.4
0.6
08
100
0.0
0.2
0.4
0.6
08
1.0
0.0
0.2
044
0.5
0.8
1.0
0.0
0.2
0.4
0.
08
1.00
1.00
1.00
1.00
1:02
0.75
0.75
0.75
0.75
0.73
055-
0:25
0.00
00 0.6 0.6
08
0.00
0.0
02
04
10
0.00
0.0
0.2
04
0.6
10
0.00
0.0
02
0.4
0,5
0.00
0.0
04
02
Mcon Startztic GT
Mean Statstic GT
Maan Statitc GT
Mcan Statistic GT
Mcan Statetic GT
Figure 18: Calibration of perfect predictor (predicting Statistic-GT) for different ESP test cases.
26

---

Published as a conference paper at ICLR 2025
F.3 GENERALIZABILITT OF ECE'S DIAGONAL BEHAVIOR
F.3.1
OTHER DNN ARCHITECTURES
AR-NCA
multi- Jayer ConvLSTM
ConvL5TTM-atk-2
Convi.S Sorvi-STM-btlk-d
5.000
0.000
5.011
8,003
6:106
6.002
0.040
8.004
0.023
093
0.005
ELITT
Bublic
E.doo
924
6:905
050
124
4. 6.0222
6:001
0:038
0:07+
122
0/007
6:00
0.804
(a) ECE
AR-NCA
multi-layar ConVLSTM
ConviS STM-b-k---
2.004
0L550
0.340
8.14R
C.370
014
0.641
0.380
0.00
6.1T
3.687
0,899
0,000
6.123
8.328
0.35
0.625
933
0.324
0,152
(b) MSE
AR-NCA
multti-aayor COnVLSTM
ConvL.5TMM-bt-22
ConviL STM-bbk-dd
00.99
0.817
6.000
115
0.094
0.810
0.000
5.006
9.30G
0.333
0.114
0.004
0.239
0.230
0,255
6:277
6:301
323
0.810
0,582
0.1XX
0,225
0.054
6:073
3300
0.000
0.50X
0.000
2:333
0-0.771
0.937
6.551
40
0.378
0.500
(c) 1 - AUC-PR
Figure 19: Evaluation metric scores for four different DNNs, from strongest (leftmost) to weakest (rightmost) using evaluation metrics (a) ECE, (b) MSE, (c) AUC-PR, and (d) Recall. While MSE, AUC-PR, Recall depict marked degradation in performance, ECE remains fairly stable, though slight increase is observed. In general, ECE shows lower variation on the synthetic dataset, indicating that it has a lower discriminating power.
Details about the DNN Architectures: (1) AR-NCA Kang al. 2024): AR-NCA involves recurrent cellular attention module that couples LSTM and cellular self-attention, (2) multi-layer ConvL (num laum_layrrs=2), (3) convLSTM-btk-2 convLSTM with spatial bottleneck that downsamples by 2, (4) convLSTM-bik- convLSTM with a spatial bottleneck that downsamples by 4. Using the deterministic case as the basis (train-test s-level=0), DNN quality is of the order 1 R 2 4.
F.3.2 OTHER EVALUATION METRICS
BCE
2.10
5.18
ENERGY SCORE
CORRELATION
8.20
20 po.0..00. 0.666 0224
0.122
6336
3:345
0:349
0.303
0.135
20
08-99993
8,033
0:937
0.330
15500.12 0,154 0.144
0.246
0,376
1411
0.12
1.000
4.76
0.39
71X
505
0.73
0.06
3,803
2.356
3.43
2.93
6.30
0.02
8.007
8.630
0.009
0.117
6.376
C33009
4:25
6.500
3.49
Test tovel
S-leval
S-leval
(8)
(b)
(c)
(d)
Figure 20: Matrix heatmap for error- based scores. First three, BCE, CRPS, Energy Score are proper scoring rules. Last one, is popular spatial correlation based evaluation metric. None of them exhibit the sensitivity to stochasticity which can be observed for ECE.
F.4 LONG HORIZON BEHAVIOR OF ECE VS. BASELINES
Figure 22 illustrates the DNN performance for the forest fire model using AUC-PR, MSE, and ECE when the s-levels of the training and test data match, corresponding the diagonal in Figure
27

---

Published as a conference paper at ICLR 2025
AUC
1- RECALL
6.31
PACCBION
20 0.ass 5,00
0.037
0.022
BALL
0.713
155 o.ss 0.000
0:034
0.020
0.022 0.00
0.024
0.02
0.022
0.072 6.040 0.022
6457
0.00
0.50
0.924
0.072
6.773
6.802 9,00
0.033
8,002
0.030
0.033 9AIZ 6.037 0,014 8.010
39
035
049
665
0:763
0.899
082
0.030 9,077 0.009 0.004 0.002
mmh
903
0.00
0.48
Test Nevel
Text S-lvel
(8)
(b)
(c)
(d)
Figure 21: Matrix heatmap for other classification based metrics.
S-Level
10
15
20
0.16-
2.
0.25
0.14-
018
0.12-
0.10-
00
10-
Mean predicted value
0.00
Step
Figure 22: Long Horizon Performance of the DNN using (a) AUC-PR, (c) MSE, and (d) ECE [calibration curve in inset]. ECE demonstrates that the DNN's long horizon predictions remain stable.
3
Each evaluation score aggregates predictions and GT across 300 test simulations per time step. While baselines show declining scores with increasing S-Level and longer prediction horizons, ECE scores remain stable, indicating faithfulness to the Statistic- G.T. In the special case of deterministic evolution (S-Level 0), all three evaluation metrics agree. For perfect predictor, Refinement 0 for deterministic evolution since frac [mm) 0 or 1 ($3.4.2) resulting in stable MSE. AUC-PR (F2R) is effective since there only one possible realization. AUC-PR is measuring the ability the DNN to generalize over different forest and fire seed configurations considering deterministic evolution.
F.5
ECE CONVERGENCE NUMBER OF SAMPLES
DNN trained tested same S-Level
DNN trained on S-Level
0.08
S.aevo
0.30
Salave
+
0.06
0.2-
05
0.10
0.00
0.00
Number
Number Samples
Figure 23: Testing the asymptotic guarantees of ECE. Each sample is grid of predictions (64 64) at 1=59 in the test split of the synthetic forest dataset. As the number of samples increase in the calculation, ECE converges. We can observe, that for the asymptotic guarantees to kick in, sufficient number of samples required.
28

---

Published as a conference paper at ICLR 2025
G APPLICATION OF DEEP LEARNING IN WILDFIRE MODELING
G.1 MODELS FOR WILDFIRE MODELING
Currently, forest fire spread models are categorized into three classes: empirical, semi-empirical. and physical. Empirical models, e.g., DNN modeling using remote sensing data (Jain et al. 2020). analyze fire data statistically without exploring combustion mechanisms. Semi-empirical models, often the preferred choice, like (Finney 1998 Rothermel 1972) integrate physical laws, such as heat transfer, but necessitate resource-intensive ground surveys for calculating model parameters (Finney 1998). Physical models, involving complex equations for heat dynamics (Séro- Guillaume & Margerit 2002), are too complex for broad application. All these classes of models are deterministic and do not explicitly assume fire dynamics to be stochastic.
Table 5: Current Evaluation Metries
Description
Proportion of true posilive
Table 4: Selected works in Wildfire Prediction
Precision
predictions among all positive predictions
Proportion of true posttive
Work
Window Size
DNN
Evaluation Metrle
Recall
predictions arnong actual positive instances
Proportion of correct
Radke et al.
2019
Prod.: T424h
Obs.: T
CNN
Fl-score,
Recall, Accurccy
Accuracy
predictions among predictions
Harmonic mean of precision rocall
Yang et al.
2021
Obs: [I:52w, T]
CNN.
AUC-ROC, MSE
Fl-score
Prod.: T45w
LSTM
Area under the precision-recalll curve, evaluating trade between precision and recall
2022
Obs: T
CNN
Precision, Recall
AUC-PR,
AUC-PR
Area under receiver operating characterishe curve, evaluating trade-off between true positive rate and false posirive rate
AUC-ROC
Mcan squared error, measuring avorage squared differeace between predicted and actual values
MSE
DNN-based Modeling of Wildfires. Wildfire prediction a critical application area, driven by the increasing frequency and severity of wildfires (Westerling et al. [0066] Thompson et al. 2016). DNNs weather, topology. and historical fire data- to predict future fire maps, identifying burn locations (Huot et al. 2022 Radke et al. 2019 Yang et al. 2021). These models learn the stochastic "rules" of fire evolution to improve prediction accuracy. Leveraging the broad spatial and temporal coverage of satellite (Huot et al.) 2022) and aircraft-based sensor data (Doshi et al. 2019). DNNs capture fire dynamics with reduced operational costs compared to conventional tools like FARSITE (Finney 1998), which rely on ground-based data.
utilize multivariate
ceervvationsiincuudee
The growing application of DNNs in wildfire prediction evident in studies summarized in Table 4 and reviewed comprehensively in (Jain et al. 2020). For instance, FireCast (Radke et al.) 2019) improves 24-hour wildfire perimeter prediction accuracy by 20% using satellite imagery (Finney 1998). These models integrate covariates such vegetation, terrain, and weather; for example, Huot et al. use a 64 x 64-pixel grid containing observational variables (Huot et al., 2022).
DNN-based wildfire prediction operates in two phases: learning fire evolution rules from observations and applying this knowledge for future predictions. Models are trained using Binary Cross Entropy (BCE) loss and evaluated with metrics summarized in Table 5
PAUC-ROC is recommended due class imbalance
Huot al.
2022 Sofaer
2019a
29

---

Published as a conference paper at ICLR 2025
G.2
OBSERVATIONAL VARIABLES IN NEXT DAY WILDFIRE SPREAD (NDWS) DATASET
Table 6: Observational variables and their description in the NDWS dataset (Huot et al. 2022)
Observational Variable
Description
Elevation
Terrain height above sea level
Wind Direction
The direction from which the wind originates
Wind Speed
Velocity of the wind
Minimum Temperature
Lowest daily temperature
Maximum Temperature
Highest daily temperature
Humidity
Amount of water vapor in the air Amount of rain, snow, etc., that falls
Precipitation
Drought Index
Measure of dryness indicating drought conditions Vegetation indices indicating plant health and coverage Number of individuals per unit area Indicator of fire potential energy release
Vegetation
Population Density
Energy Release Component (ERC)
G.3 CALIBRATION CURVE ON NDWS DATASET
1.0
ECE 0.01245
Fraction of positives
0.8
Frequency
2
0.6
0.6
w
0.2
0.4
0.. 0.2
0.9
8.0
011
0.2
0.3
04
0.5
0.6
0.7
038
0.9
0.0
0.4
0.8 1.0
DNN
Figure 24: [Left] displays the histogram of forecasts generated by Conv-AE: [Right] shows Calibration Curve that illustrates forecast accuracy in an interpretable manner. This curve suggests (1) the DNN's tendency towards overconfidence in mid-range forecasts, and (2) the accuracy of its probabilistic predictions is better for forecasts at the lower and upper ends of the probability spectrum.
G.4 RESOLVING RANK CONFLICT IN DNN RANKINGS
Table 7: Performance of DNNs on NDWS Dataset (ranks in parentheses by each evaluation metric)
DNN
AUC-PR+
MSEJ
ECE!
Conv-AE (Hut et al.
2022
0.2473 (5)
0.0124 (4)
0.0119 (4)
Conv-CA
0.2631 (4)
0.0146 (5)
0.0207 (5)
AR-NCA (Kag et al.
2024
0.2790 (2)
0.0099 (2)
0.0012 (1)
SegFormer (Xie et al. 202T
0.2727 (3)
0.0100 (3)
0.0020 (2)
U-Net (Ronneherger
2015
0.3302 (1)
0.0096 (1)
0.0023 (3)
Conflicts in model rankings are key to model selection (Heaton et al. 2018). We simulate selecting
the optimal DNN and explore how ECE complements existing metrics.
DNN Models used. We use five architectures: Conv-AE, convolutional autoencoder from Huot et al. (Huot et al. 2022); Conv-CA, a modified Conv-AE without the spatial bottleneck (details in Appendix E. ; AR-NCA, an Attentive Recurrent Neural Cellular Automata for locally interacting discrete systems like forest fires (Kang et al. 2024);; SegFormer, a transformer- based segmentation model (Xie et al., 202T); and U-Net, convolution based model (Ronneberger et al.) 2015).
Results and Discussion. preesenss evaluation scores (AUC-PR, MSE, and ECE) for five DNNs on the test split, revealing rank conflicts between the metrics. This raises the question: Which metric
30

---

Published as a conference paper at ICLR 2025
best reflects model performance? Based findings, each metric evaluates different system property: AUC-PR measures fidelity to the Observed-GT, ECE assesses fidelity to the Statistic-GT, and MSE lies somewhere in between. In complex systems, failure to replicate the Observed-GT isn't necessarily problematic due to inherent stochasticity, but a high ECE indicates fundamental mismatch between the trained DNN and the test data, meaning the model should not be used. The evaluation strategy should prioritize ECE as the first metric, as a high ECE indicates the DNN has failed to capture the test data's stochasticity. Afterward, fidelity realization metries should be applied. This study ultimately advocates for multi-faceter evaluation approach, with fidelity to the stochastic process as crucial factor. While the NDWS dataset has one-step forecast horizon, limiting our ability to assess ECE's s stability over time, future work with datasets featuring longer horizons could explore this further.
31

---

Published as a conference paper at ICLR 2025
H A PRACTICAL USER GUIDE FOR EVALUATING COMPLEX SYSTEMS
Evaluation Framework
MSE
MSE
MSE
F2SP
ECE
MSE
ECE
ECE
ECE
AUC-PR
1-AUC-PR (A)
1 AUC-CP
1 AUC-PP
F2R
(B)
(C)
Figure 25: Visualization of prediction performance across different scenarios within the proposed evaluation framework. The evaluation framework distinguishes between two criteria: F2R and F2SP. F2R evaluates how closely the model predictions match single observed outcome, assessed using metrics like AUC-PR (plotted along the x-axis as 1 - AUC-PR, where lower values are better). F2SP evaluates whether the model captures the underlying stochastic process, measured by ECE (y-axis, where lower values indicate better calibration). The dashed gray line represents MSE, which balances F2R and F2SP. The symbols represent the DNN's performance for different metrics. (A) Low errors across all metrics indicate the DNN matches the Observed-GTT and captures the stochastic process. (B) High AUC-PR error but low ECE suggests the DNN has learned the stochastic process but fails to replicate the single Observed-GT. (C) High errors across all metrics suggest the DNN fails to match both the Observed-(T) and the stochastic process. This visualization helps users analyze model performance across different evaluation criteria. See Table 8 for detailed interpretation.
Complex systems are modeled interactions, resulting in highly divergent outcomes and posing unique challenges for evaluation. This guide introduces practical evaluation framework for machine learning (or statistical) models that forecast the behavior of such high-dimensional complex systems. The framework evaluates model predictions against ground truth while accounting for the inherent randomness in system evolution.
of interacting components. These components exhibit random
Traditional evaluation criteria focus matching single observed outcome. However, in stochastic systems, this outcome represents just one sample of the underlying process, as identical starting points can yield diverse results. This makes the current evaluation criteria inadequate.
Our evaluation framework offers a unique perspective on model performance in stochastic complex systems by distinguishing between Fidelity to Realization (F2R), which focuses matching single observed outcome, and Fidelity to Stochastic Process (F2SP), which evaluates whether a model captures the system's broader stochastic dynamics ($3). Using metrics like AUC-PR, ECE, and MSE, the framework provides a cohesive evaluation strategy by integrating these metrics within a unified structure. Rather than treating them independently and risking rank conflicts ($G.4), this framework aligns the metrics to offer comprehensive and actionable understanding of model's ability to balance individual outcome prediction with capturing the underlying stochastic process. Figure visually summarizes this framework, highlighting different scenarios based on these metrics.
Examples of Applications:
Remote Sensing Grids: Cells represent land areas, e.g., wildfire spread or weather events.
Virtual Grids: Cells represent assets in financial markets or road segments in traffic systems.
Epidemiology: Cells track infection in regions or neighborhoods.
The next page provides detailed explanation of the framework's implementation and interpretation.
32

---

Published as a conference paper at ICLR 2025
FRAMEWORK REQUIREMENTS
Modeling Complex Systems on a Grid: We model the system on a grid of size H x W, where:
Grid Cells: Each cell (i,j) at time occupies one of m states, 82,(6.5) E ; 8m)
Target State: A specific state s* is tracked over time for each grid cell.
Observed Ground Truth (Observed-GT)): For state s", define (1,(6.5) = 1 if (1,(1.) s*,
else O. Collectively, these values form the ground truth grid B, (bt, H
DNN Predictions: For each grid cell, the DNN predicts Pe.(t.1) E [0. 1]. representing the
probability of s*. These predictions form the grid-level output P. = (1,)) Hx which "'x
is compared against the corresponding B, over the forecast horizon 1.,....
CALCULATING EVALUATION METRICS
For each time step calculate evaluation metrics using ground truth B, and predictions P. (standard library implementations in footnotes):
AUC-PR: Measures how well predictions match the Observed-GT (F2R
ECE: Assesses calibration against expected outcomes (F2SP
MSE: Balances F2R and F2SF
Note: Ensure the grid size H x W is large enough for reliable metric computation.
Larger grids or batch sizes improve reliability (see limitations of ECE in $7
Example Usage: In our evaluation, we tested 300 simulations where the DNN observed the first 10 frames and predicted the next 50. This resulted in 300 50 x 64 x 64 (B T x H W) ground truth samples (B₂) and predictions (Pt), each of size 64 x 64. For the
plot in Figure 4
scores were calculated for each time step I along the B axis. For the stochastic matrix in Figure 3 these scores were averaged across time into a single value. Metrics can also be computed in batches for finer-grained analysis, provided sufficiently large batch size is used for convergence (see FF.5
INTERPRETING EVALUATION SCENARIOS WITH THE PROPOSED FRAMEWORK
Table 8 summarizes three evaluation scenarios to help users interpret results and their implications.
Table 8: Evaluation Scenarios and Their Impact.
Scenario 1 AUC-PR
ECE
Interpretation and Impact
A. Low Errors Low
Low
The DNN matches the Observed-GFT and is well- calibrated. capturing both the specific outcome and the underlying stochastic process, Ideal for low-stochasticity environments, this indicates the model is suitable for deployment.
Across All Metrics
B. High AUC-PR Er- High ror but Low ECE
Low
The DNN is well-calibrated but doesn't 't match the Observed-GFT. It captures the stochastic process but fails to replicate single realization. making it useful for high-stochasticity settings longer prediction horizons. Predictions provide valuable insights into risk and variability.
C. High Errors Across All Metries
High
High
The DNN is poorly calibrated and fails to match the Observed-Git indicating fundamental issues with model training system complexity. Predictions unreliable and unsuitable for decision- making.
Scikit-lcarn AUC documentation
Incertainty-calibre documentation
Scikit-leam MSE documentation
33


Reviews:
[{"review_id": "H9PaE0SeEj", "reviewer": "Reviewer_cBqL", "summary": "the metric of expected calibration error is introduced and studied as a way to capture fidelity of a learned representation to an underlying stochastic process (rather than a single realization of that process, as with typical metrics like AUC or MSE).", "questions": "- I don't understand the second part of the critical question, \"is it encountering different stochastic behaviours\" (different from what)? how is the \"differentness\" relevant?\n- While it's pretty clear to me how to use this immediately in my work, I think anyone who wasn't already aware they wanted exactly this might struggle. Could you provide something like a \"practical users guide\" for non-domain experts?\n - if the clarity of the plots can be improved, the naming of the stat/metric you're introducing, and improve it's \"usability\" to the community, I would be happy to upgrade my score. You've done great work and this would bring the paper to the level it deserves.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "Great paper, wonderfully practical and insightful; I've been looking for something like this for 5+ years! Nice eval on real-world data.\nI started writing a thing I would like to you add and then discovered it was already in the paper (long horizon behaviour)", "weaknesses": "While overall the paper is very clear, some of the captions and explanations of the experiments/insights from them and how they tie to the figures could be improved. \nSome specifics:\n- first fig should say what you mean by realization, and F2R and F2SP should be bolded (not ital) to make them easy to find in the text. Observed GT should be explained a bit more, or maybe it would be enough to move the sentence currently after F2R to be the second sentence of the paragraph.\n - Fig 5 is unclear to me. What is the data, what is S-level, why is it \"good\" that the 20 vs 10 lines are far apart? All of this should be clear from the caption\n - the clarity wanes a bit as the paper goes on, and it's a bit confusing that you call it ECE vs. F2SP vs Statistic-GP. Do these different namings really serve something? It could be a lot more clear if you just have one naming.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LJFxtjxdPS", "reviewer": "Reviewer_7Rfe", "summary": "This paper presents a study evaluating deep neural networks (DNNs) within stochastic complex systems, emphasizing the importance of Expected Calibration Error (ECE) in measuring fidelity to stochastic processes. The findings are validated through multiple experiments and comparisons.", "questions": "A question arises regarding Figure 1: Can ECE be an effective metric for measuring F2R compared to other available metrics? Figure 1 suggests that the answer may be *no*.\n\nAn important indicator that ECE is a reliable measure is its diagonal pattern, showing low scores only when training and test S-Levels align, as illustrated in Figure 4. Could the authors provide theoretical insights to support this indicator?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The topic of evaluating DNNs within stochastic complex systems is both intriguing and important.\n\nIn the primary evaluations, the author conducted experiments across various settings, including different DNN architectures, comparisons with multiple evaluation metrics, and diverse simulation tasks.\n\nThe main text clearly explains the difference between ECE in classical assessment and stochastic process settings.", "weaknesses": "The paper is somewhat difficult to follow. For example, providing a brief introduction to the structure of each section would enhance clarity, particularly in Sections 2 and 3. Additionally, it is difficult to grasp the main messages conveyed by the table in Figure 2(b). Furthermore, in lines 229–240, the macro-level concept is introduced abruptly, which may disrupt the clarity and readability of the main text.\n\nThe main findings' practical applicability appears limited. In real-world scenarios, data generally provides only a single observed outcome centered on observable ground truth (line 117). Since the primary evaluation is simulation-based, the controlled stochasticity falls short of capturing real-world complexity. The Statistic-GT is basically derived by normalizing the frequency of target state occurrences across multiple Monte Carlo simulations.\n\nMinors:\n\nM1. The original text for the abbreviation RV is not given.\n\nM2. In Table 1, what about the possibility of recovery in the Host-Pathogen problem?\n\nM3. In line 152, maybe consider using an alternative symbol for Moore neighborhood, instead of $\\mathcal{N}$ (normally representing Gaussian distribution).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wFirUKJfZc", "reviewer": "Reviewer_sNm1", "summary": "This paper presents a study on evaluating deep neural networks designed to forecast the evolution of stochastic complex systems. The authors identify a gap in traditional evaluation methods—such as threshold-based classification metrics and error-based scoring rules—which focus on a model's ability to replicate observed ground truth but fail to assess how well the model has learned the underlying stochastic process. To address this issue, they introduce a new property called Fidelity to Stochastic Process, representing the DNN's ability to predict the statistical ground truth of the stochastic process.\n\nThe paper proposes using the Expected Calibration Error (ECE) as an evaluation metric that satisfies the necessary conditions for assessing fidelity to statistical ground truth. This work underscores the importance of capturing the underlying stochastic processes in deep neural networks  evaluations for complex systems.", "questions": "L50: Is --> is (lowercase)\nFig1: no need to write the whole name, you can use acronyms because they're already defined in the text, however MSE is not defined at this point.\nL88: fidelity to realization --> F2R (it was already defined previously, so you can use the acronym)\nL99: the notation of the dimension of the real vector O_t is confusing, what is (R^n)^(H x W), is n = H x W? If so, make that explicit.\nTable 1: some rows end with full stop, other don't. Please make it consistent. Either all with or all without.\nI find it odd to place Figures in columns as Figure 1 (which has a large top white margin) and Figure 3. I would suggest column figures into one row figure with multiple subfigures as you did with Figure 2. \nL201: Isn't the indicator variable already defined as B_t in L99? Why defining again with different notation?\nL298: MSE already defined in text previously, no need to write the whole name again.\nL516: ECE already defined in text previously, no need to write the whole name again.\nTable 2 and Table 7: highlight the best performing DNNs.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "The paper makes a significant contribution by introducing the concept of Fidelity to Stochastic Process (F2SP), a novel evaluation criterion specifically designed to assess a DNN's ability to learn the underlying stochastic interactions in complex systems.\n\nThe authors provide a rigorous formalization of F2SP within a stochastic framework, establishing clear criteria for its valid measurement. The use of Expected Calibration Error (ECE) as an evaluation metric is well-justified.", "weaknesses": "I found it hard to read the paper because there was a lack of consistency in the acronyms, the authors would redefine them in several parts of the text again and again. I addressed my comments on text in the questions section. \n\nIn the tables, the best neural networks based on each criterion are not highlighted, which makes it difficult to the reader to infer and correlate the arguments in the text. I addressed my comments on text in the questions section. \n\nThe focus of the paper is primarily on binary or discrete prediction tasks, leaving out regression tasks where the definition of calibration is more complex. While the authors acknowledge this and suggest it as an area for future work, the current scope limits the immediate applicability of the findings to a broader range of problems involving continuous outcomes.\n\nAdditionally, the use of the NDWS dataset, which is restricted to next-day predictions, prevents the assessment of ECE over longer time horizons, which are common in many complex systems. Could you elaborate on how future work might address this limitation? \n\nThe paper highlights the lack of open-source complex system datasets as a barrier to broader validation. Are there any ongoing initiatives or plans to develop, collect, or standardize such datasets?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Eyz3NpsTvi", "reviewer": "Reviewer_HXsj", "summary": "This work offers a new perspective on evaluating DNNs in stochastic complex systems by emphasizing  the importance of capturing underlying the stochastic process. Traditional evaluation methods assess the DNN’s ability to replicate the observed ground truth but fail to measure the DNN’s learning of the underlying stochastic process. This paper proposes a new property called Fidelity to Stochastic Process, representing the DNN’s ability to predict the ground truth of the stochastic process, and introduces an evaluation metric that exclusively assesses fidelity to  the ground truth of the stochastic process. The Expected Calibration Error is used to evaluate the fidelity to ground truth of statistic process. Empirical experiments on synthetic datasets (including wildfire, host-pathogen, and stock market models) and real-world wildfire data are used to show the measurement of fidelity to stochastic process by Expected Calibration Error.", "questions": "As mentioned in the \"Weaknesses\" part.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper offers a new perspective on evaluating DNNs by considering DNNs as stochastic processes and uses a widely used criteria in Bayesian Deep Learning application to assess the fidelity to stochastic process. This work clearly explains the Expected Calibration Error is used to assess DNN modes in three synthetic cases and one real world case.", "weaknesses": "This paper is well organized and well written, several minor issues should be addressed: (1) The explaination of figures is not sufficient, e.g., in Figure 2 (1), the label for x-axis is not specified (I guess it is time?), either add a label or explain it in the captions. Same problems also exist in Figure 4. (2) This work examines ECE on three synthetic environments (forest fire, host-pathogen and stock market models) and a real world wildfire spread dataset. I can tell that these datasets are all multivariate either for classification or regression. Maybe due to the limit of pages, the authors didn't include the experiments on images. I suggest the authors add some discussions or comments in the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2FFAFx0AgZ", "reviewer": "Reviewer_iuWX", "summary": "This paper introduces a novel stochasticity-compatible evaluation strategy for assessing existing models in the context of complex systems. The author justifies the Expected Calibration Error (ECE) as suitable for assessing the model fidelity of stochastic systems through both simulation environments and real-world data.", "questions": "I'm curious about how the author evaluates ECE at time $t$ based on Statistic-GT $P_{t}$. Do we have to simulate it again from $t=0$ for $N$ times or we can sample states from $t-1$ and go forward $N$ times (the system is Markov)? Can we still apply ECE on Statistic-GT when the system is not Markov?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. Evaluating model fidelity on the stochastic system is significant and has wide applications.\n2. The paper is well-motivated and both the dataset and experiments are thorough.", "weaknesses": "1. Although the author attempts to explain the difference between their work and ECE in deep learning in Lines 282-288, it appears to me this work is still a direct application of using ECE to evaluate the model performance on a stochastic system. The author is encouraged to discuss more in-depth about the distinction between ECE in the proposed method (stochasticity comes from evolving in the environment, aka, Statistic-GT) and ECE in previous works (stochasticity comes from the output distribution).\n2. In Lines 243-244, the author claims that Statistic-GT is more stable than classification-based metrics, but I could not find any evidence related to calculating ECE on Statistic-GT is less sensitive to the system variance than MSE. Is there any theoretical support for using ECE over MSE on stochastic systems with different noise levels and could the author clarify it a bit more?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)