Net-Zero: A Comparative Study on Neural Network Design for Climate-Economic PDEs Under Uncertainty
Carlos Rodriguez-Pardo [a,b,c,*] , Louis Daumas [a,b,c] , Leonardo Chiani [a,b,c] and Massimo Tavoni [a,b,c]
a Politecnico di Milano b Euro-Mediterranean Center on Climate Change (CMCC) c RFF-CMCC European Institute on Economics and the Environment (EIEE) ORCID (Carlos Rodriguez-Pardo): https://orcid.org/0000-0001-6121-7738, ORCID (Louis Daumas): https://orcid.org/0000-0003-3239-8538, ORCID (Leonardo Chiani): https://orcid.org/0009-0007-2491-6290, ORCID (Massimo Tavoni): https://orcid.org/0000-0001-5069-4707
Abstract. Climate-economic modeling under uncertainty presents significant computational challenges that may limit policymakers’ ability to address climate change effectively. This paper explores neural network-based approaches for solving high-dimensional optimal control problems arising from models thatincorporate ambiguity aversion in climate mitigation decisions. We develop a continuoustime endogenous-growth economic model that accounts for multiple mitigation pathways, including emission-free capital and carbon intensity reductions. Given the inherent complexity and high dimensionality of these models, traditional numerical methods become computationally intractable. We benchmark several neural network architectures against finite-difference generated solutions, evaluating their ability to capture the dynamic interactions between uncertainty, technology transitions, and optimal climate policy. Our findings demonstrate that appropriate neural architecture selection significantly impacts both solution accuracy and computational efficiency when modeling climate-economic systems under uncertainty. These methodological advances enable more sophisticated modeling of climate policy decisions, allowing for better representation of technology transitions and uncertainty—critical elements for developing effective mitigation strategies in the face of climate change. 1 Introduction
Climate change represents one of humanity’s most pressing challenges, demanding robust policy responses. Integrated assessment models (IAMs) that combine climate and economic systems have become essential tools for evaluating mitigation strategies. Yet, they face significant computational and conceptual challenges when incorporating realistic uncertainty related to climate impacts and mitigation processes. These uncertainties are particularly pronounced when considering disruptive decarbonization technologies whose future availability and effectiveness are unknown. IAMs often rely on deterministic approaches or simple uncertainty representations, failing to capture the complexinterplay between decision-making under uncertainty and technological transitions. Recently, sophisticated stochastic models incorporating ambiguity aversion and model misspecification have emerged. These better rep
∗ Corresponding Author. Email: carlos.rodriguezpardo.jimenez@gmail.com. resent the challenges faced by policymakers who must act without knowing probability distributions governing climate and technological dynamics. However, the high-dimensional partial differential equations (PDEs) that characterize these models quickly become computationally intractable using numerical methods. Recent advances in deep learning provide promising directions for addressing these challenges. Neural network-based PDE solvers have demonstrated remarkable capabilities in handling high-dimensional problems that would be infeasible for traditional methods. However, their application to climate economics models with ambiguity, and discrete technological jumps remains unexplored. Different neural architectures may vary in their ability to capture the unique characteristics of these problems, such as discontinuities in policy functions around technological breakthroughs or the complex effects of uncertainty aversion on optimal control. Therefore, we argue that there is alack of understanding of the design choices that govern the performance of neural PDE solvers in climate and economic applications. This gap may lead to inaccurate solutions or hinder the possibility of achieving high-performing, scalable models in this field. Our main goal in this paper is, therefore, to explore the design space of neural PDE solvers for climate-economic modeling under uncertainty. These models face unique computational challenges including high dimensionality and non-linear interactions between economic and climate variables. Leveraging a ground-truth dataset generated using a Finite-Differences solver, we explore the impact of several important neural network design factors, such as architecture types, residual connections, activation functions, optimization algorithms, and regularization techniques. We find that commonly used approaches often provide suboptimal trade-offs between computational efficiency and accuracy, hindering progress in the field and limiting the application of neural PDE solvers for climate-economic policy design. By applying targeted modifications to these models, we demonstrate significantimprovements in both accuracy and computational efficiency. We present the following contributions:
- A continuous-time endogenous-growth economic model incorporating mitigation options under uncertainty. - A systematic evaluation of neural architectures for solving the re sulting PDEs, comparing their accuracy and computational effi ciency against finite-difference solutions. - New insights into how neural network design affect the ability to model optimal climate policy under uncertainty. - An open source implementation and data made available upon publication, enabling further research at the intersection of climate economics and machine learning. 2 Related Work
Our work intersects climate economics under uncertainty, neural approaches to solving partial differential equations (PDEs), and computational methods for climate policy analysis. We build upon key advances in these domains while addressing important limitations. Climate Economics under Uncertainty
Climate-economic models increasingly incorporate uncertainty to better represent real-world decision contexts. While early contributions used deterministic integrated assessment models (IAMs) like DICE, more recent approaches have employed stochastic extensions to account for climate risk. Beyond this, ambiguity aversion frameworks have been developed where decision-makers face unknown probability distributions. The Hansen-Sargent model misspecification framework offers a rigorous approach to handling deep uncertainty by optimizing against worst-case scenarios. Recent applications by Barnett et al. and Barnett et al. to climate economics demonstrate its impact on the social cost of carbon and financial markets. Berardi further explored how uncertainty and sentiment affect asset prices in this framework. Our work extends these models by incorporating mitigation technologies and explicitly modeling uncertain technological jumps—responding to calls from Grant et al.
about the need to better represent mitigation deterrence under uncertainty. Neural Networks as PDE Solvers
Traditional numerical methods for PDEs face significant challenges in high-dimensional settings due to the curse of dimensionality. The Physics-Informed Neural Networks framework introduced neural approaches for solving PDEs, while the Deep Galerkin Method (DGM) pioneered a mesh-free approach using domain sampling. Al-Aradi et al. extended this approach with their Policy Iteration Algorithm (DGM-PIA), which simultaneously optimizes value functions and control policies in an adversarial framework. Recent advances have explored specialized architectures for PDE solving, including periodic activations in SIREN, spectral approaches in Fourier Neural Operators, attention mechanisms, and various residual connections strategies. However, as noted by Cuomo et al., comprehensive comparisons of neural architectures for climate-economic PDEs remain scarce. Huang et al. provides a recommendable survey on the topic, but lacks applicationspecific guidance for complex economic systems. Computational Challenges in Climate Policy
The computational complexity of climate-economic models under uncertainty has limited their application in policy settings. Leading models either use simplified representations with few states or employ approximations that may miss critical interactions.
Neural PDE solvers offer promising alternatives, but their application to climate economics has focused primarily on baseline architectures. Barnett et al. demonstrated the power of neural methods for climate policy analysis but did not explore architectural variations. Anagnostou et al. surveyed machine learning for climate policy models, highlighting the need for more rigorous benchmarking. Our methodological contribution fills this gap by systematically evaluating neural architectures for climate-economic PDEs with mitigation options. Deep network architectures have also been used to tackle broader types of uncertainty. Friedl et al. developed the Deep Equilibrium Network methodology to run sensitivity analysis on key components of stochastic IAMs. Our work advances the state-of-the-art by: developing a richer climate-economic model incorporating multiple mitigation pathways and technological uncertainty; systematically evaluating neural architectures against finite-difference benchmarks; and providing insights into how architectural choices affect solution quality for climate policy applications. This approach enables more sophisticated modeling of technology transitions under uncertainty—addressing a critical need in climate policy design. 3 Overview
Our paper examines neural architecture design for climate-economic PDEs under uncertainty. We illustrate our method in Figure 1.
We structure the rest of the paper as follows. In Section 4.1, we present our economic model with multiple mitigation options and explicit uncertainty representation, deriving the resulting Hamilton-JacobiBellman equation. Next, in Section 4.2, we formulate the specific PDEs arising from this economic framework, highlighting the mathematical properties that challenge numerical solvers and motivate our architectural exploration. We describe our finite-difference implementation in Section 4.2.2, which we use to generate ground truth solutions for benchmarking. The methodological contributions appear in Section 5, where we systematically present neural network design that address the specific characteristics of our climateeconomic PDEs. Section 5 briefly outlines our optimization strategy and computational implementation, followed by Section 6 which presents our comparative analysis of architectural performance, and shows how these methodological advances translate to enhanced capabilities for analyzing climate policy questions under uncertainty, with Section 7 summarizing key insights and future directions. 4 Methods
In this section, we formalize the climate-economic model we will use for our experiments, our finite-differences solution, and how we compute the benchmark metrics. 4.1 Economic model
For our climate-economic model, we consider a modified version of the continuous-time stochastic economic framework proposed by
. We choose this model for its relative simplicity, which makes it feasible to compute solutions using traditional Finite Differences methods, key for robust benchmarking of neural solvers. This model is composed of four states variables: k, the logarithm of the capital stock ( K ), S L, the share of low-carbon capital in the economy, Γ the temperature anomaly, and n the logarithm of climate damage intensity ( N ). We assume two controls, i Land i H, the investment rates in low and high-carbon capital respectively. This problem is solved assuming that a benevolent social planner with rational expectations DGM Variants SIREN
Residual Networks FNO
Recurrent Networks Self-Attention
Optimal Architecture Value Network Policy Network
Economic Model
- Multiple mitigation options
- Ambiguity aversion
Evaluation
- Value and policy errors
- Computational efficiency
HJB Equation
- Value function
- Optimal control
- Uncertainty adjustments
Ground Truth Finite-Difference Solution
Figure 1 : Our neural PDE method for climate-economic models under uncertainty. Our approach builds on a climate-economic model, formulated as a Hamilton-Jacobi-Bellman equation.
We compare various neural architectures within a two-network framework against finitedifference ground truth solutions. needed) maximizes welfare given by:
∞
V = � 0
e [−][ρ] log( C ) dt . 0
∞ ∞
e [−][ρ] u ( C ) dt = 0 � 0
motion of the log-value of capital j . Note that as per Îtô’s Lemma, the laws of motion in level would write:
dKL =][ +] 2 [(][σ] [L] [S L] [)][σ] [L] [W] [L] [,]
In the equation, ρ is the discount rate and Cis consumption given by:
C = Y − I,
with Y being the production and I the investment. Y is given by Y = α K . A fraction S L of the capital K is made of low-carbon capital, such that emissions are given by:
E = λα K,
where the parameter λ is the carbon intensity.
We define the following shorthands:
dk [∗] = dk − S L σ L W L +σ H W H,
dS [∗] L [=] [ dS] [ L] [−] 2 [S] [ L][ −] [S] [ L] [)(][σ] [L] [W] [L] [ +][ σ] [H] [W] [H] [)][,]
d Γ [∗] = d Γ − σ Γ dW,
dn [∗] = dn −σ Γ dW,
and write the HJB problem as:
ρ V = ρ log � α K − i LS L N − i H � +
12 ∂∂ kV �(σ LS L ) +) [�] +
∂ V
∂∂ Vk [dk] [∗] [+] ∂ [∂] S [V]
∂ S L [dS] [ ∗] L [+]
1 ∂ V 2 ∂ S L
�)] � +
∂∂ V Γ [d] [Γ] [∗] [+] 2 V [E] [σ] Γ [+]
∂ V
2 ∂∂ nV [] [E] [σ] Γ [.]
∂∂ Vn [dn] [∗] [+] 2
2 Again, considering K = KL + K H = KL
KL [K] [ +] [K] K [H] This expression can be further simplified by verifying that V = v − n. As a result, the problem is reduced to a three-state HJB, in which [∂] [V] [=][ −] [∂] [V] [=]
First-order conditions We first consider the set of previous controls and derive the marginal value of consumption:
mc = ρ [∂] [lo][g(] [C] [)]
∂ C = [(][ℓ][)] ρ
[∂] [V] [∂] [V]
∂ n [=][ −] ∂ n
∂ n [=]
(α K − S Li [(] L [ℓ][)] [K] [−][−] [S] [ L] [)] [i] [(] H [ℓ][)] [K] [.]
ρ v = ρ log � α K − i LS L N − i H �
We then derive the first-order conditions for the two controls, that determine their optimal values given the set of gradients: � i [] � ∗ = 2κ1 L �1 − ∂ V [(][ℓ][)] − mc − [∂] [V] [(][ℓ][)] �,
∂ V
∂ S L [dS] [ ∗] L [+]
∂∂ Vk [dk] [∗] [+] ∂ [∂] S [V]
12 ∂∂ kV �(σ LS L ) +) [�] +
∂ S L
�1 − ∂ V ∂ [(] k [ℓ][)] −] − [∂] ∂ [V] [(][ℓ][)]
1 ∂ V 2 ∂ S L
�)] � +
, �
∂ S L
∂∂ V Γ [d] [Γ] [∗] [+] 2 V [E] [σ] Γ [−] [dn] [∗] [.]
This gives us a partial-differential equation that we will aim to approximate with various neural network architectures. 4.2 Finite-di ff erence approach to PDEs
A standard approach to PDE solving is the use of finite-difference methods. We solve this problem with this more traditional approach to provide ground truth to evaluate our neural networks. We use the same method as.Considering the three-state problem defined above, we define a linear grid along the three states k, S Land Γ, with points separated by spaces h k, h S Land h Γ . For each point of the grid, we compute the gradient of the value function, derive controls, and update the value function accordingly by solving for it with a linear solver. We iterate these operations until convergence. Our algorithm is summarized as follows, and we detail the steps below. Input : Initial guess for value function v 0, ϵ = 1 e [−]
Initialize n = 0, v [n] 0 [=] [ v] while | v [n] 0 [+] − v [n] 0 [|][ > ϵ] [ do] Step 1: Solve for optimal controls { i [n] L [+] [,] [ i] [n] H [+] [}] Shallow approach applied Step 2: Solve for value function Linear solver (implicit scheme) applied Step 3: Check for convergence if | v [n] 0 [+] − v [n] 0 [|][ < ϵ] [ then] [ stop, otherwise continue] end if end while
Return v [n] 0
4.2.1 Policy iterations
Gradient computation To derive the controls, we first compute the gradient of our value function along the grid. For interior points, we use a central-difference scheme with natural boundary conditions on the edges of the domain. For a state x, withi the indexation of state values along the grids, h x the corresponding grid step and ℓ the iteration index: � ∂ V ∂ x [(][ℓ][)] � ( i ℓ) = f − h x f,
� i [] � ∗ = 2κ1 H
 1 − ∂ V ∂ k [(][ℓ][)] − mcS
V ∂ k [(][ℓ][)] − S L [∂] ∂ [V] S [(][ℓ][)]
. For stability, we update the controls with a relaxation parameter χ:
i [] = χ � i [] � ∗ + i ( L ℓ) [,]
i [] = χ � i [] � ∗ + i ( H ℓ) [.]
We update the value of consumption and of the laws of motion and proceed to the linear solver. We also implemented a Cobweb algorithm which, instead of directly updating the controls, starts with a first guess on the control and updates the marginal value of consumption, iterating until convergence.
No differences were found across solutions, we thus proceed with the shallower approach. 4.2.2 Linear solver
We compact the HJB equation as:
∂ [V] k [B] [k] [+] ∂ [∂] S [V]
[∂] [V] [∂] [V]
∂Γ [B] [Γ] [+] ∂ k
∂ [∂] k [V] [C] [k] [+] ∂ [∂] S [V]
[∂] [V] [∂] [V]
∂ S L [C] [S L] [+] ∂Γ
∂Γ [C] [Γ] [+] [D] [ =][,]
− A ρ v + [∂] [V]
[∂] [V] [∂] [V]
∂ S L [B] [S L] [+] ∂Γ
V [(][ℓ][)] (ℓ)
∂ x � ii = f − h x f
∂ xi = + 2 h x −,
� ∂ ∂ Vx[(][ℓ][)] � ( i ℓ) = f + fh −2 f ( i ) . V [(][ℓ][)] (ℓ)
∂ x � i
−
h x . = f + f −2 f ( i ) i h
Central difference schemes are widely used given their stabilising property. For a lower-boundary point, we use a forward difference:
∂ V [(][ℓ][)] (ℓ) f − f � ∂ x � 0 [=] h x,
V [(][ℓ][)] (ℓ)
∂ x � 0
f − f 0 [=] h x
∂ x 0 [=] h x,
∂ V [(][ℓ][)] (ℓ) f + f −2 f � ∂ x � 0 [=] h . V [(][ℓ][)] (ℓ)
∂ x � 0
h x . f + f −2 f 0 [=] h
For a higher-boundary point, we use a backward difference:
∂ V [(][ℓ][)] (ℓ) f − f � ∂ x � I −1 [=] h x,
V [(][ℓ][)] (ℓ)
∂ x � I −
f − f I −1 [=] h x
− −
∂ x I −1 [=] h x,
∂ V [(][ℓ][)] (ℓ) f + f −2 f � ∂ x � I −1 [=] h . ∂ V [(][ℓ][)] (ℓ)
∂ x � I −
− −
h x . f + f −2 f I −1 [=] h
With A = II, and where II is the identity matrix.
The first-order partial derivative coefficients are:
Bk = S L] − κ L] ) +
1 2 [(][σ] [L] [S] [ L] [)] [+]
] − κ H] ) +
1 2 [)] [,]
BS L = S L] − κ L] ) +
1 2 [(][σ] [L] [S L] [)] [−]
] − κ H] ) +
1 2 [)] [))][.]
B Γ = E ζ
The second order partial derivative coefficients are:
Ck = 2 �(σ LS L ) +) [�],
CS L = 2 �)] �,
C Γ = 2 [E] [σ] Γ [.]
The constant term is given by:
D = ρ log α K − S Li L K − i H K � �
− d Γ + 2 [η] [σ] Γ [E] [�] . �
The linear solver is builtin Eigen, and uses an implicit scheme for better performances. For better accuracy, we derive the finitedifference solution over a 60 grid, and with a tolerance of 10 [−] . Around 1500 iterations are needed to reach a solution, which takes approximately 2 hours to converge. Architecture Key Advantages Potential Strengths for Climate-Economic PDEs DGM Variants Foundational approach with proven success in PDE solving; Effective baseline for HJB equations; explicitly designed to mesh-free sampling enables scaling to higher dimensions handle PDE residuals with non-uniform boundary
DGM Variants Foundational approach with proven success in PDE solving; Effective baseline for HJB equations; explicitly designed to mesh-free sampling enables scaling to higher dimensions handle PDE residuals with non-uniform boundary conditions
Residual Networks Improved gradient flow in deep networks; allows for very Stable training across high-dimensional spaces; better preserdeep architectures without vanishing gradients vation of low-level features important for policy decisions
Residual Networks Improved gradient flow in deep networks; allows for very Stable training across high-dimensional spaces; better preserdeep architectures without vanishing gradients vation of low-level features important for policy decisions
Highway Networks Adaptive information flow control; dynamically decides Better handling of varying sensitivities across state variables; whichinformation passes through the network can selectively focus on critical state regions
Highway Networks Adaptive information flow control; dynamically decides Better handling of varying sensitivities across state variables; whichinformation passes through the network can selectively focus on critical state regions
Recurrent (LSTM/GRU) Captures dependencies between variables; maintains internal May better model state-dependent transitions memory through recurrent connections
May better model state-dependent transitions
SIREN High-frequency component modeling; periodic activations capture oscillatory behavior
SIREN High-frequency component modeling; periodic activations Represents sharp transitions around technological jumps; betcapture oscillatory behavior ter approximation of discontinuities in policy functions
FNO Global pattern recognition; operates efficiently in frequency Model interactions across the full state space; captures multidomain; learns spatial relationships scale dynamics between climate and economy
FNO Global pattern recognition; operates efficiently in frequency Model interactions across the full state space; captures multidomain; learns spatial relationships scale dynamics between climate and economy
Self-Attention Dynamic relationship modeling; context-dependent feature Capturing climate-economic variable interactions; adaptively weighting focuses on relevant state information
Self-Attention Dynamic relationship modeling; context-dependent feature Capturing climate-economic variable interactions; adaptively weighting focuses on relevant state information
Table 1 : Comparison of Neural Network Architectures for Climate-Economic PDE Solving. Validation against Ground Truth Base Architecture: Deep Galerkin Method
Base Architecture: Deep Galerkin Method
This process provides us with Ground Truth, which will consist of three outcomes: the value function V [∗∗] and the controls i [∗∗] L [and] [ i] [∗∗] H [.] More precisely, our adversarial training framework will involve two deep networks. One V approximates the value function, while the other, P, returns the controls, that we denote ι Land ι H . To evaluate our architectures, we evaluate our trained models on the 60 grid. We measure the geometric mean of absolute errors across the grid G:
~~[�]~~ 3 L final = Li V × Li L × Li H,
Our foundation is the Deep Galerkin Method (DGM) introduced by Sirignano & Spiliopoulos. The DGM approach employs deep neural networks trained to satisfy PDEs through stochastic sampling of the domain. A key advantage of this approachis its mesh-free nature, which helps overcome the curse of dimensionality inherentin our climate-economic model. We systematically explore several variants of the base DGM-PIA architecture:
1. DGM Baseline : The original DGM architecture with TanH activations and non-residual complex hidden layers, matching the original implementation as closely as possible. 2. DGM with SiLU : Replaces TanH with Sigmoid Linear Units activation functions, defined as:
SiLU( x ) = x - σ( x ),
where σ( x ) is the logistic sigmoid function. These activations provide smoother gradients, whichis particularly beneficial around the discontinuities caused by technological jumps in our model. 3. DGM with SiLU and Residual Connections : Enhances the
SiLU-based DGM with residual connections between layers:
hi +1 = hi + f i ( hi ),
where hiis the output of layer i and f iis a nonlinear transformation. Residual connections facilitate gradient flow during training, which enhances stability and fast convergence. Alternative Architecture Paradigms
Beyond enhancing DGM variants, we explore fundamentally different architectural paradigms that may capture distinct mathematical properties of our climate-economic PDEs. For these models, we also employ SiLU activations where appropriate, and residual connections in every layer f i . We implement and test the following models:
1. Residual and Highway Networks :
- Residual Networks: Uses standard residual connections with normalized activations, where each layer f i works as a shallow multi-layer perceptron. This architecture helps maintain gradient flow in very deep networks.
3. SIREN (Sinusoidal Representation Networks): Employs periodic activations:
sin. This architecture may better represent solutions with highfrequency components, which can emerge in our economicclimate system near transition boundaries. 4. FNO (Fourier Neural Operator): Implements a spectral approach that leverages the Fourier domain:
h [′] = FFT [−] ( R (θ) · FFT( h )),
where R (θ) represents learnable filters in the Fourier domain. This approachis particularly suited for capturing both global and highfrequency patterns in the PDE solution. 5. Multi-Head Self-Attention Mechanisms: Incorporates transformer-style attention:
Attention( Q, K, V ) = softmax � Q ~~√~~ Kd [T]
2. Adaptive Optimization Strategies : We test multiple optimization algorithms with varying learning rates for value and policy networks. 3. Learning Rate Scheduling : We test step scheduling and cosine annealing schedules with optional warm restarts to navigate the complex loss landscape. 4. Regularization and Stabilization : We explore dropout, weight decay, and gradient clipping to improve training stability. Hyperparameter and Training Details
To ease the navigation of the complex design space of these adversarial framework, we design a comprehensive hyperparameter optimization, with the goal of finding combinations of hyperparameters that provide satisfactory solutions, thereby minimizing the distance with respect to the finite-differences solution L final . In terms of Optimization, we observe that AdamW with high weight decay, batch size of 4096, a gradient norm clipping of 1.0, and betas of β 1 = 0.9, β 2 = 0.99 provide satisfactory results. Learning rates should be different for each model, and that learning rate schedulers do not provide consistentimprovements. For each batch of samples, we train P for 20 steps, and V also for 20. Sampling is done uniformly across the space. In terms of model sizes, regardless of the layer design, we find that both models provide the best results when they have 5 hidden layers each, where the layers of P have 1024 neurons each, and V has 512. Dropoutis set to 0 as it failed to provide consistent gains. Note that this specific configuration works generally well for every combination of V and P, butit may not be necessarily optimal. Itis, however, computationally unfeasible to test every possible combination of model design, size, optimization type, regularization, and every other hyperparameter choice. Our goal in this paper is to provide insights on this design space (with a particular focus on layer designs), rather than exploring it exhaustively. We train the models for 500 epochs, and run validation against the ground truth data at the end. In the following, we will provide experiments on how the architecture design impacts the final performance of our neural PDE solvers, using the training configuration we just described.
~~√~~ d
d
V . �
These layers excel at capturing complex relationships between variables of different characteristics, potentially helping model the interactions between economic and climate dynamics. Architecture-Specific Enhancements
We implement several specialized techniques to address particular challenges in our climate-economic model:
1. Boundary-Enforcing Constraints : Custom output layers with sigmoid activations enforce known boundary constraints on policy functions, particularly for investment rates that must remain within physically meaningful bounds. Figure 3 : Comparison on the share of low-carbon capital confidence ranges), emulated by our model on different prices assumptions. 6 Results
Neural Network Comparisons Our evaluation of neural architectures for climate-economic PDEs reveals several key insights into the relationship between network design, solution quality, and computational cost. The error analysis shows a clear performance stratification across architectures. MLP-based networks consistently achieve the lowest error rates when combined when MLP or recurrent models, outperforming specialized alternatives. This counter-intuitive finding suggests that the smoothness properties of climate-economic solution manifolds may be better captured by the universal approximation capabilities of MLPs than by the inductive biases of specialized architectures. Particularly striking is the poor performance of SIREN and Self-Attention networks, despite their theoretical advantages in representing high-frequency and complex components—indicating that such components may play a less significant role in our specific HJB formulation than anticipated. Interestingly, baseline DGM models can be improved with minor architectural adjustments, including the use of SiLU activations (DGM+S) and residual connections (DGM+SR), although these results are somewhat marginal. Models that operate on a Fourier space (FNO) generally provide good results as Value Networks, althoughinferior than recurrent layers like LSTM or GRU. Our best-performing method is trained on 16 minutes on a RTX 4090 GPU. Using CodeCarbon, we measure an electricity consumption of 0.127646 kWh. Computational efficiency measurements reveal that architecture choice creates performance differences of up to 6× in training time, with MLPs demonstrating a clear advantage. The combined error-efficiency metric highlights the practical superiority of MLPMLP configurations, achieving considerably better efficiency than attention-based alternatives. This suggests that the computational complexity of climate policy models might be effectively managed without resorting to more sophisticated architectures. Notably, our results challenge conventional practices regarding architecture selection for PDE solving. The significant performance gap between theoretical expectations and empirical results for specialized architectures (e.g., Self-Attention and SIREN) reveals a potential misalignment between these networks’ inductive biases and the specific mathematical properties of climate-economic PDEs. The relatively smooth, low-frequency nature of optimal policy functions in our model appears to favor the simplicity and trainability of MLPs
over architectures designed for more complex systems. Our findings also demonstrate the asymmetric impact of value and policy network architecture choices. Value networks appear to have a greater influence on overall performance, with MLP value networks producing superior results regardless of policy network selection. This asymmetry provides practical guidance for computational resource allocation when designing neural PDE solvers for climate policy applications. These results show that careful design choices for the value function are more relevant than those for the policy model, which are comparably easier to train. Model Emulation Once trained, we can use our model to test the impact of different hypothesis on the structure of the economy and its potential decarbonization. On Figure 3, we show an emulation of our best performing model for the years 2020-2050, under different assumptions on the cost of low carbon capital. As shown, on more expensive scenarios, low carbon capital becomes a smaller share of the economy, thus hindering decarbonization efforts. These emulations also provide uncertainty estimates, and can enable more fine-grained policy designs, as they work on a fully continuous
space. 7 Conclusions
This paper has explored the impact of neural network architecture design on solving high-dimensional partial differential equations arising in climate-economic models. Our systematic evaluation of neural architectures demonstrates that design choices significantly affect both solution accuracy and computational efficiency when modeling climate policy under uncertainty. We examined various neural architectures including enhanced DGM variants, MLPs, RNN, SIREN, FNO, and attention-based architectures. Our analysis reveals that architecture selection plays an important role in capturing the unique characteristics of climate-economic PDEs. We also discovered that pairing different architectures for value and policy networks often yielded better results than using identical architectures for both components, whichis the common practice. Our findings show that simpler neural networks offer better results bothin accuracy and computational cost for this problem, paving the way for more scalable and performing neural PDE solvers in climate economics. These methodological advances enable more sophisticated modeling of climate mitigation pathways under uncertainty, allowing policymakers to better understand the complexinterplay between ambiguity aversion, technological transitions, and optimal policy design. Our approach helps bridge the gap between theoretical model complexity and practical policy analysis, by providing more accurate and computationally efficient solutions to high-dimensional climateeconomic PDEs. Future work should extend our quantitative evaluation to additional model types and optimization algorithms, as well as explore higher-dimensional problems thatincorporate additional mitigation technologies such as carbon capture.