Numerical methods

The optimization of mitigation scenarios requires libraries that allow for: assembly and integration of numerous heterogeneous models; scaling with high-dimensional variables, due primarily to the disagregation among periods, technologies, and markets; minimal extra implementation when incorporating or modifying models.

Multidisciplinary Optimization (MDO) frameworks are classically used to optimize or improve a given design under various constraints [93], it offers practical methods to handle model integration and coupling. Efficient optimization under high-dimension can also be achieved by using gradient-based algorithms, but their use often implies an extra burden due to the manual implementation of the derivatives of objective and constraints with regards to optimization variables. GEMSEO is an open-source Python software to automate multidisciplinary processes [94], which provided the following features that were used in the present study: automatic handling of coupled derivatives, automatic assembly of complex MDO processes based on dependency graphs, interfaces to optimization algorithms, results visualization, data storage.

Differential Programming, on the other hand, is commonly used for machine learning and scientific computing research. This programming paradigm allows for the Automatic Differentiation (AD) of lines of code. The use of AD for MDO can significantly reduce implementation burden associated to efficient high-dimension optimization. JAX is a library for array-oriented numerical computation, with capabilities for AD and just-in-time (JIT) compilation [95], enabling high-performance scientific computing in multiple hardware configurations (CPU, GPU, and TPU).

GEMSEO-JAX [96] is an open-source plug-in that was developped by the authors to bridge JAX programs into a GEMSEO process.

Computational gains

Table 6 Benchmark over a single scenario computation and linearization. Performed using a mean of 50 repetitions in a single Intel(R) Core(TM) i7-10850H CPU core with 2.70 GHz.

Method	Computation (s)	Linearization (s)
Standard (FD)	372.7	362.6
JAX version	0.45	0.48
Speedup	830	749

To evaluate the simulation gains offered by a Differential Programming paradigm, we first compare execution times over full scenario optimizations, then over a single computation and linearization.

Table 7 Benchmark over a Drop-in policy optimization with trend formulation. Performed using a single Intel(R) Core(TM) i7-10850H CPU core with 2.70 GHz, as the mean of 3 repetitions, varying technology assumptions. The objective (cumulative CO2 emissions) is normalized by the 3% of the 2°C carbon budget.

Method	Algorithm	Gradient-based	Iterations	Total time (s)	Speedup	Objective value
Standard (FD)	SLSQP	Yes	X	X	X	X
FD + JIT	SLSQP	Yes	91	69.3	87	1.72
JAX version	SLSQP	Yes	86	53.2	114	1.72
Standard	COBYLA	No	1852	6052		1.75
JAX version	COBYLA	No	1898	51	119	1.75

In Table 7, entire scenario optimization is performed with varying optimization algorithm, differentiation strategies, and compilation. Gradient-based algorithms not only require far fewer iterations for convergence but also achieve higher objective precision, confirming their advantages for high-dimensional problems [97]. Using an uncompiled model with standard Finite Differences (FD) was prohibitive due to memory demands; JIT compilation makes FD feasible yet still less efficient—both in iteration count and runtime—than forward-mode AD. Overall, using JAX yields speedups of up to two orders of magnitude for both algorithms tested.

In Table 6 the computation compares compares how long one single scenario computation takes to complete with (JAX version) and without (standard) JIT compilation, and the time required to linearize objectives and constraints. JAX introduces an initial overhead of roughly 38 s due to compilation; this is excluded from single computation comparisons but included in full-optimization timings. Once compiled, JAX achieves speedups of up to three orders of magnitude.

For context, a single scenario simulation using a restricted subset of AeroMAPS models [20] (default bottom-up setup without offsetting, non-CO2, or cost models) takes 1.29 s, about three times slower than the present approach. This comparison should be interpreted cautiously, as the modeling scopes differ substantially: AeroMAPS includes freight, non-CO2 effects, and cost models, but omits aircraft design routines and links between traffic growth and socioeconomic drivers.