In my research on exoplanets, I have heard many people talk "forward modeling of exoplanet atmospheres". I don't know what the "forward" means in "forward modeling" and how it compares with "reverse modeling", if that's even a thing.
What is forward modeling, and why is it so special that it needs to be distinguished from just plain ol' regular modeling?
There are different ways to model something. From what you're asking, there are two main types of modeling: forward modeling and inverse modeling.
Forward Modeling
In this type of modeling, you have a specific model that defines the "current" state of your system. In the case of exoplanet atmospheres, it'd likely be something that defines the molecular content, ionization level, density, etc. of your exoplanet atmosphere. Then, you use the known physics/math of your system to decide how it will behave. In this setup, what you've created is a system for predicting system states from a predetermined physics model.
Such an example would be someone creating their own atmosphere of an exoplanet in a model and then saying, okay what happens when I shine light through this atmosphere. What observations might I record?
Inverse Modeling
In some sense this is the opposite of forward modeling, albeit it doesn't really mean you're running a model to see into the past. Instead, what happens with this setup is you know a particular state or result, and you want to construct a model of your system which can produce said state. Essentially, you want your model to arrive at a certain state when it is done calculating. If it does, you have a reasonable confidence that your model was some indication of what your system is actually like.
In this situation, you'd measure components of the atmosphere, e.g. the radius of the planet as a function of wavelength, and then create a model of the atmosphere which can hopefully reproduce your observations. If you can, then the hope is that the model accurately represents what your system is.
Forward modeling is the use of a model in order to simulate an outcome. The problem of getting the model to produce data from the input is called the forward problem.
The forward model takes certain parameters and produces data that can then be compared with the actual observations.
Forward modeling seems to be in common use in the Earth sciences, referring e. g. to models of global climate, seismic events, etc.
Forward problem (direct problem, normal problem): The problem of calculating what should be observed for a particular model, e.g. calculating the gravity anomaly that would be observed for a given model of a salt dome. (A Dictionary of Earth Sciences)
The opposite procedure is called the inverse problem:
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in computer tomography, source reconstructing in acoustics, or calculating the density of the Earth from measurements of its gravity field.
It is called an inverse problem because it starts with the results and then calculates the causes. This is the inverse of a forward problem, which starts with the causes and then calculates the results.
Solving an inverse problem then means, given a set of observations, constructing a model that accounts for them.
I suppose it's to be expected that exoplanet atmospheres are studied through forward modeling, because we already have adequate atmospheric models for Earth and the understanding to adjust them to other planets, while we don't yet have an adequate characterization of exoplanet atmospheres.
From math's point of view it is simple. In linear algebra, for both, the model is the same, says $A$. Then: $$y = Ax$$
where $y$ the observation, and $x$ the physical parameters.
Forward modeling: Given $x$, calculate $y$. This is straightforward.
Inverse modeling: Given $y$, estimate $x$. Usually it is considered hard, because $A$ might be a fat matrix (more cols than rows; that said, more unknowns than number of equations), and hence hard for inversion.
The reason why forward modeling is important, is that if you solve the inverse problem using, say iterative solvers, then for each step you need at least to calculate the primal matrix-vector product ($Ax$). So when it comes to inverse modeling, forward modeling is always important (so that you know how to forward modeling for $Ax$).
Inverse modelling is where you use features of your data to estimate a set of underlying parameters of your physical model of what is going on.
Forward modelling is where you use your model to predict what you would observe and use a comparison of these predictions to your data to infer your model parameters.
A simple exoplanet example. Consider a sparsely sampled radial velocity curve. You could fit a sinusoid (or an elliptical orbit solution) to these data and estimate the period, radial velocity amplitude and then deduce a minimum mass for the orbiting exoplanet by plugging these numbers, along with an estimate of the stellar mass into the mass function formula.
A forward modelling approach would start with the mass of the star and planet, specify an orbital period and inclination and then predict what would be observed - including if necessary, functions that allow for imperfections and uncertainties in the measurements. Many such models are produced and compared with the observations until one can estimate probability functions for each of the model parameters.
To see the difference between forward and inverse models, consider our understanding that an atom can absorb and emit only certain discrete wavelengths of light. This is what we observe; we can build a simple (inverse) model of atomic structure based on these observations. But only after we had a well-developed model of the atom, such as quantum theory, were we able to predict the absorption and emission of any atom.
Forward modeling is based on these well-developed understandings and is generally the most useful form of modeling.
However, inverse models are important when we don't yet have a good understanding of a system; in that case, ad hoc models may ultimately lead us to develop entirely new models and understandings -- as was the case in understanding atoms and molecules before quantum theory was fully developed.
